WILD PFISTER FORMS OVER HENSELIAN FIELDS, K-THEORY, AND CONIC DIVISION ALGEBRAS

SKIP GARIBALDI AND HOLGER P. PETERSSON

Abstract. The epicenter of this paper concerns Pfister quadratic forms over a field F with a Henselian discrete valuation. All characteristics are considered but we focus on the most complicated case where the residue field has characteristic 2 but F does not. We also prove results about round quadratic forms, composition algebras, generalizations of composition algebras we call conic algebras, and central simple associative symbol algebras. Finally we give relationships between these objects and Kato’s filtration on the Milnor K-groups of F.

Introduction

The theory of quadratic forms over a field F with a Henselian discrete valuation is well understood in case the residue field \overline{F} has characteristic different from 2 thanks to Springer [46]. But when \overline{F} is imperfect of characteristic 2, the theorems are much more complicated—see, e.g., [50] and [22]—reflecting perhaps the wellknown fact that quadratic forms are not determined by valuation-theoretic data, as illustrated below in Example 11.14. However, more can be said when one focuses on Pfister quadratic forms over F and more generally round forms, see Part II below.

In Part III we change our focus to Kato’s filtration on the mod-p Milnor Ktheory of a Henselian discretely valued field of characteristic zero where the residue field has characteristic p. (Again, if \overline{F} has characteristic different from p, the mod-p Galois cohomology and Milnor K-theory of F are easily described in terms of \overline{F}, see [18, pp. 17–19] and [19, 7.1.10].) We give translations between valuation-theoretic properties of Pfister forms, octonion algebras, central simple associative algebras of prime degree (really, symbol algebras), and cyclic field extensions of prime degree over F on the one hand and properties of the corresponding symbols in Milnor K-theory on the other.

Along the way, we prove some results that are of independent interest, which we now highlight. Part I treats quadratic forms and composition algebras over arbitrary fields. It includes a Skolem-Noether Theorem for purely inseparable subfields of composition algebras (Th. 5.7) and a result on factoring quadratic forms (Prop. 3.12). We also give a new family of examples of what we call conic division algebras, which are roughly speaking division algebras where every element satisfies a polynomial of degree 2, see Example 6.6. More precisely, we show that—contrary to what is known, e.g., over the reals—a Pfister quadratic form of characteristic 2 is anisotropic if and only if it is the norm of such a division algebra (Cor. 6.5).

2000 Mathematics Subject Classification. Primary 17A75; secondary 11E04, 16W60, 17A45, 19D45, 19F15.

Part II focuses on round quadratic forms and composition algebras over a field F with a 2-Henselian discrete valuation with residue field of characteristic 2. From this part, our Local Norm Theorem 8.10 has already been applied in [56]. We also relate Tignol’s height ω from [51] with Saltman’s level hgt_{com} from [45], with a nonassociative version of Saltman’s level that we denote by hgt_{ass}, and with valuation-theoretic properties of composition algebras, see Th. 12.11 and Cor. 19.3. We show that composition division algebras over F having pre-assigned valuation data, subject to a few obvious constraints, always exist (Cor. 11.13), though they are far from unique up to isomorphism (Example 11.14).

Part III gives a K-theoretic proof of the Local Norm Theorem (Th. 15.2); it has an easy proof but stronger hypotheses than the version in Part II. Finally, our Gathering Lemma 16.1 is independent of the rest of the paper and says that one may rewrite symbols in a convenient form.

Let us now discuss what happens “under the hood”. The basic technical result in Part I is a non-orthogonal analogue of the classical Cayley-Dickson construction for algebras of degree 2. It is used to prove the Skolem-Noether Theorem mentioned above as well as to construct the examples of conic division algebras.

In Part II, we proceed to a more arithmetic set-up by considering a base field F, discretely valued by a normalized discrete valuation λ: F → ℤ ∪ {∞}, which is 2-Henselian in the sense that it satisfies Hensel’s lemma for quadratic polynomials. Our main goal is to understand composition algebras and Pfister quadratic forms over F. For this purpose, the results of [41]—where the base field was assumed to be complete rather than Henselian—carry over to this more general (and also more natural) setting virtually unchanged; we use them here here without further ado. Moreover, we will be mostly concerned with “wild” composition algebras over F (see 7.15 below for the precise definition of this term in a more general context) since a complete description of the “tame” ones in terms of data living over the residue field of F has been given in [41]. (Alternatively—and from a different perspective—one has a good description of the tame part of the Witt group of F from [16].) The approach adopted here owes much to the work of Kato [23, § 1], Saltman [45] and particularly Tignol [51] on wild associative division algebras of degree the residual characteristic p > 0 of their (possibly non-discrete) Henselian base field. Moreover, our approach is not confined to composition algebras but, at least to a certain extent, works more generally for (non-singular) pointed quadratic spaces that are round (e.g., Pfister) and anisotropic. We attach valuation data to these spaces, among which not so much the usual ones (ramification index (7.9 (b)) and pointed quadratic residue space (7.9 (c))), but wildness-detecting invariants like the trace exponent (8.1) play a significant role. After imitating the quadratic defect [35, 63A] for round and anisotropic pointed quadratic spaces (8.8) and extending the local square theorem [35, 63:1] to this more general setting (Thm. 8.10), we proceed to investigate the behavior of our valuation data when passing from a wild, round and anisotropic pointed quadratic space P having ramification index 1 as input to the output Q := ⟨⟨μ⟩⟩ ⊗ P, for any non-zero scalar μ ∈ F (Section 9). In all cases except one, the output, assuming it is anisotropic, will again be a wild pointed quadratic space. Remarkably, the description of the exceptional case (Thm. 9.9), where the input is assumed to be Pfister and the output turns out to be tame, when specialized to composition algebras, relies critically on the nonorthogonal Cayley-Dickson construction encountered in the first part of the paper (Cor. 10.18). This connection is due to the fact that our approach also lends itself to the study of what we call λ-normed and λ-valued conic algebras (Section 10), the latter forming a class of conic division algebras over F that generalize ordinary composition algebras and turn out to exist in all dimensions 2^{n}, n = 0, 1, 2, …, once F has been chosen appropriately (Examples 10.7,10.15). There is yet another unusual feature of the exceptional case: though exclusively belonging to the theory of quadratic forms (albeit in an arithmetic setting), it can be resolved here only by appealing to elementary properties of flexible conic algebras (Thm. 10.17). The second part of the paper concludes with extending Tignol’s notion of height [51], which agrees with Saltman’s notion of level [45], to composition division algebras over F and relating them to the valuation data introduced before (Thm. 12.11).

In the third part of the paper, we consider the case where F has characteristic zero and a primitive p-th root of unity and has a Henselian discrete valuation with residue field of characteristic p. In that setting, Kato, Bloch, and Gabber gave a description of the mod-p Milnor K-groups k_{q}(F) in [23, 24, 25, 4]; we use [10] as a convenient reference. We relate properties of a symbol in k_{q}(F) with valuationtheoretic properties of the corresponding algebra, see Prop. 19.1 and Th. 19.2.

Contents

Introduction 1

Part I. Base fields of characteristic 2 4 1. Standard properties of conic algebras 4 2. Flexible and alternative conic algebras. 8 3. Interlude: Pfister bilinear and quadratic forms in characteristic 2 12 4. A non-orthogonal Cayley-Dickson construction 16 5. The Skolem-Noether theorem for inseparable subfields 20 6. Conic division algebras in characteristic 2. 25

Part II. 2-Henselian base fields 27 7. Pointed quadratic spaces over 2-Henselian fields. 27 8. Trace and norm exponent. 33 9. Valuation data under enlargements. 38 10. λ-normed and λ-valued conic algebras. 45 11. Applications to composition algebras. 52 12. Types of composition algebras and heights. 59

Part III. Connections with K-theory 66 13. Introduction to Part III 66 14. The filtration on K-theory 67 15. The Local Norm Theorem 8.10 revisited 68 16. Gathering the depth 70 17. Ramification index for symbols 72 18. Theorem 9.9 revisited 73 19. Dictionary between K-theory and algebras and quadratic forms 74 20. Proof of Proposition 19.1: case q = 1 75 21. Proof of Proposition 19.1: cases q = 2 77 22. Proofs of Theorem 19.2 and Corollary 19.3 78

References 79

Part I. Base fields of characteristic 2

1. Standard properties of conic algebras

Although composition algebras are our main concern in this paper, quite a few of our results remain valid under far less restrictive conditions. The appropriate framework for some of these conditions is provided by the category of conic algebras. They are the subject of the present section.

We begin by fixing some terminological and notational conventions about nonassociative algebras in general and about quadratic forms. For the time being, we let k be a field of arbitrary characteristic. Only later on (Sections 3−6) will we confine ourselves to base fields of characteristic 2.

1.1. Algebras. Non-associative (= not necessarily associative) algebras play a dominant role in the present investigation. For brevity, they will often be referred to simply as algebras (over k) or as k-algebras. A good reference for the standard vocabulary is [57].

Left and right multiplication of a k-algebra A will be denoted by x ↦ L_{x} and x ↦ R_{x}, respectively. A is called unital if it has an identity (or unit) element, denoted by 1_{A}. A subalgebra of A is called unital if it contains the identity element of A. Algebra homomorphisms are called unital if they preserve identity elements. Commutator and associator of A will be denoted by [x, y] = xy − yx and [x, y, z] = (xy)z − x(yz), respectively. If A is unital, then

Nuc(A) := {x ∈ A | [A, A, x] = [A, x, A] = [x, A, A] = {0}}

is a unital associative subalgebra of A, called its nucleus, and

Cent(A) := {x ∈ Nuc(A) | [A, x] = {0}}

is a unital commutative associative subalgebra of A, called its centre. We say A is central (resp. has trivial nucleus) if Cent(A) = k1_{A} (resp. Nuc(A) = k1_{A}).

A k-algebra A is called flexible if it satisfies the flexible law

xyx := (xy)x = x(yx). (1)

A is said to be alternative if the associator is an alternating (trilinear) function of its arguments. This means that A is flexible and satisfies the left and right alternative laws

x(xy) = x^{2}y, (yx)x = yx^{2}. (2)

Furthermore, the left, middle and right Moufang identities

x(y(xz)) = (xyx)z, x(yz)x = (xy)(zx), ((zx)y)x = z(xyx) (3)

hold.

1.2. Quadratic forms. Our main reference in this paper for the algebraic theory of quadratic forms is [15], although our notation will occasionally be different and we sometimes work in infinite dimensions. Let V be vector space over k, possibly infinite-dimensional. Deviating from the notation used in [15], we write the polar form of a quadratic form q: V → k, also called the bilinear form associated with q or its bilinearization, as ∂q, so ∂q: V × V → k is the symmetric bilinear form given by

∂q(x, y) := q(x + y) − q(x) − q(y)  (x, y ∈ V).

Most of the time we simplify notation and write q(x, y) := ∂q(x, y) if there is no danger of confusion. The quadratic form q is said to be non-singular if it has finite dimension and its polar form is non-degenerate in the usual sense, i.e., for any x ∈ V, the relations ∂q(x, y) = 0 for all y ∈ V imply x = 0. Recall that nonsingular quadratic forms have even dimension if the characteristic is 2 [15, Chap. 2, Remark 7.22].

1.3. Conic algebras. We consider a class of non-associative algebras that most authors refer to as quadratic [57] or algebras of degree 2 [33]. In order to avoid confusion with Bourbaki’s notion of a quadratic algebra [5], we adopt a different terminology. A k-algebra C is said to be conic if it has an identity element 1_{C} ≠ 0 and there exists a quadratic form n: C → k with x^{2} − t(x)x + n(x)1_{C} = 0 for all x ∈ C, where t is defined by t := ∂n(1_{C}, −): C → k and hence is a linear form. The quadratic form n is uniquely determined by these conditions and is called the norm of C, written as n_{C}. We call t_{C} := t = ∂n_{C}(1_{C}, −) the trace of C and have

(1)  x^{2} − t_{C}(x)x + n_{C}(x)1_{C} = 0, n_{C}(1_{C}) = 1, t_{C}(1_{C}) = 2  (x ∈ C).

Finally, the linear map

ι_{C}: C ⟶ C, x ⟼ ι_{C}(x) := x^{∗} := t_{C}(x)1_{C} − x, (2)

called the conjugation of C, has period 2 and is characterized by the condition

(3)  1_{C}^{∗} = 1_{C}, xx^{∗} = n_{C}(x)1_{C}  (x ∈ C).

The property of an algebra to be conic is inherited by unital subalgebras. Injective unital homomorphisms of conic algebras are automatically norm preserving. A conic algebra C over k is said to be non-degenerate if the polar form ∂n_{C} has this property. Thus finite-dimensional conic algebras are non-degenerate iff their norms are non-singular as quadratic forms. Orthogonal complementation in C always refers to ∂n_{C}. We say C is simple as an algebra with conjugation if only the trivial (two-sided) ideals I ⊆ C satisfy I^{∗} = I.

1.4. Invertibility in conic algebras. Let C be a conic algebra over k. By (1.3.1), the unital subalgebra of C generated by an element a ∈ C, written as k[a], is commutative associative and spanned by 1_{C}, a as a vector space over k; in particular it has dimension at most 2. We say a is invertible in C if this is so in k[a], i.e., if there exists an element a^{−1} ∈ k[a] (necessarily unique and called the inverse of a in C) such that aa^{−1} = 1_{C}. For a to be invertible in C it is necessary and sufficient that n_{C}(a) ≠ 0, in which case a^{−1} = n_{C}(a)^{−1}a^{∗}. The set of invertible elements in C will always be denoted by C^{×}.

As usual, a non-associative k-algebra A is called a division algebra if for all a, b ∈ A, a ≠ 0, the equations ax = b, ya = b can be solved uniquely in A. The quest for conic division algebras is an important topic in the present investigation. The following necessary criterion, though trivial, turns out to be useful.

1.5. Proposition. The norm of a conic division algebra is anisotropic. □

The converse of this proposition does not hold (cf. 6.1). For char(k) ≠ 2, conditions that are necessary and sufficient for a conic algebra to be division have been given by Osborn [36, Thm. 3].

1.6. Inseparable field extensions. Exotic examples of conic algebras arise in connection with inseparability. Suppose k has characteristic 2 and let K/k be a purely inseparable field extension of exponent at most 1, so K^{2} ⊆ k. Then K is a conic k-algebra with n_{K}(u) = u^{2} for all u ∈ K, ∂n_{K} = 0, t_{K} = 0 and ι_{K} = 1_{K}. In particular, inseparable field extensions of exponent at most 1 over k are degenerate, hence singular, conic division algebras.

1.7. Composition algebras. Composition algebras form the most important class of conic algebras. Convenient references, including base fields of characteristic 2, are [29, 47], although [29] introduces a slightly more general notion. An algebra C over k is said to be a composition algebra if it is non-zero, contains a unit element and carries a non-singular quadratic form n: C → k that permits composition: n(xy) = n(x)n(y) for all x, y ∈ C. Composition algebras are automatically conic. In fact, the only quadratic form on C permitting composition is the norm of C in its capacity as a conic algebra.

1.8. Basic properties of composition algebras. Composition algebras exist only in dimensions 1, 2, 4, 8 and are alternative. They are associative iff their dimension is at most 4, and commutative iff their dimension is at most 2. The base field is a composition algebra if and only if it has characteristic different from 2. Composition algebras of dimension 2 are the same as quadratic étale algebras. Composition algebras of dimension 4 (resp. 8) are called quaternion (resp. octonion or Cayley) algebras. Two composition algebras are isomorphic if and only if their norms are isometric (as quadratic forms). The conjugation of a composition algebra C over k is an algebra involution.

What is denied to arbitrary conic algebras holds true for composition algebras:

1.9. Norm criterion for division algebras. A composition algebra C over k is a division algebra if and only if its norm is anisotropic. Otherwise its norm is hyperbolic, in which case we say C is split. Up to isomorphism, split composition algebras are uniquely determined by their dimension, and their structure is explicitly known.

1.10. The Cayley-Dickson construction. The main tool for dealing with conic algebras in general and composition algebras in particular is the Cayley-Dickson construction. Its inputs are a conic algebra B and a non-zero scalar μ ∈ k. Its output is a conic algebra C := Cay(B, μ) that is given on the vector space direct sum C = B ⊕ Bj of two copies of B by the multiplication

(u_{1} + v_{1}j)(u_{2} + v_{2}j) := (u_{1}u_{2} + μv_{2}^{∗}v_{1}) + (v_{2}u_{1} + v_{1}u_{2}^{∗})j (u_{i}, v_{i} ∈ B, i = 1, 2). (1)

Norm, polarized norm, trace and conjugation of C are related to the corresponding data of B by the formulas

(2)  n_{C}(u + vj) = n_{B}(u) − μn_{B}(v),
(3)  n_{C}(u_{1} + v_{1}j, u_{2} + v_{2}j) = n_{B}(u_{1}, u_{2}) − μn_{B}(v_{1}, v_{2}),
(4)  t_{C}(u + vj) = t_{B}(u),
(5)  (u + vj)^{∗} = u^{∗} − vj

for all u, v, u_{i}, v_{i} ∈ B, i = 1, 2. Note that B embeds into C as a unital conic subalgebra through the first summand; we always identify B ⊆ C accordingly. The Cayley-Dickson construction Cay(B, μ) is clearly functorial in B, under injective unital homomorphisms.

It is a basic fact that C is a composition algebra iff B is an associative composition algebra. Conversely, we have the following embedding property, which fails for arbitrary conic algebras, cf. Example 10.8 below.

1.11. Embedding property. Any proper composition subalgebra B of a composition algebra C over k is associative and admits a scalar μ ∈ k^{×} such that the inclusion B ↪ C extends to an embedding Cay(B, μ) → C of conic algebras. More precisely, μ ∈ k^{×} satisfies this condition iff μ = −n_{C}(y) for some y ∈ B^{⊥} ∩ C^{×}.

1.12. The Cayley-Dickson process. Let B be a conic k-algebra. Using non-zero scalars μ_{1}, …, μ_{n} ∈ k^{×} (n ≥ 1), we write inductively

C := Cay(B; μ_{1}, …, μ_{n}) := Cay(Cay(B; μ_{1}, …, μ_{n−1}), μ_{n})

for the corresponding iterated Cayley-Dickson construction starting from B. It is a conic k-algebra of dimension 2^{n}dim_{k}(B). We say C arises from B and the μ_{1}, …, μ_{n} by means of the Cayley-Dickson process. The norm of C is given by

n_{C} = ⟨⟨μ_{1}, …, μ_{n}⟩⟩ ⊗ n_{B}. (1)

Here are the most important special cases of the Cayley-Dickson process.

Case 1. B = k, char(k) ≠ 2. Then n_{C} = ⟨⟨μ_{1}, …, μ_{n}⟩⟩ is an n-Pfister quadratic form. C is a composition algebra iff n ≤ 3.

Case 2. B = k, char(k) = 2. Then n_{C} = ⟨⟨μ_{1}, …, μ_{n}⟩⟩_{q} is a quasi-Pfister (quadratic) form [15, § 10, p. 56]. Moreover, n_{C} is anisotropic iff C = k(\sqrt{μ_{1}}, …, \sqrt{μ_{n}}) is an extension field of k, necessarily purely inseparable of exponent 1, hence never a composition algebra.

Case 3. B is a quadratic étale k-algebra. Then n_{B} = ⟨⟨μ⟧ for some μ ∈ k [15, Example 9.4] and (1) shows that

n_{C} = ⟨⟨μ_{1}, …, μ_{n}, μ⟧

is an (n + 1)-Pfister quadratic form over k. Moreover, C is a composition algebra iff n ≤ 2.

Composition algebras other than the base field itself always contain quadratic étale subalgebras. Hence, by Cases 1, 3 above and by the embedding property 1.11, they may all be obtained from each one of these, even from the base field itself if the characteristic is not 2, by the Cayley-Dickson process. The preceding discussion also shows that all Pfister and all quasi-Pfister quadratic forms are the norms of appropriate conic algebras.

1.13. Inseparable subfields. Let C be a composition division algebra over k. A unital subalgebra of C is either a composition (division) algebra itself or an inseparable extension field of k; in the latter case, k has characteristic 2 and the extension is purely inseparable of exponent at most 1 [53]. The extent to which this case actually occurs may be described somewhat more generally as follows.

Suppose k has characteristic 2, B is a conic k-algebra and μ_{1}, …, μ_{n} ∈ k^{×} are such that the norm of

C := Cay(B; μ_{1}, …, μ_{n}) (1)

is anisotropic, so all non-zero elements of C are invertible (1.4). Then, by Case 2 of 1.12,

K := Cay(k; μ_{1}, …, μ_{n}) ⊆ C

is a purely inseparable subfield of degree 2^{n} and exponent 1.

Specializing this observation to n = 2 and B quadratic étale over k, we conclude that every octonion division algebra over a field of characteristic 2 contains an inseparable subfield of degree 4.

2. Flexible and alternative conic algebras.

This section is devoted to some elementary properties of flexible and alternative conic algebras. In particular, we derive expansion formulas for the norm of commutators and associators that turn out to be especially useful in subsequent applications.

Phrased with appropriate care, most of the results obtained here remain valid over any commutative associative ring of scalars. For simplicity, however, we continue to work over a field k of arbitrary characteristic. We fix a conic algebra C over k and occasionally adopt the abbreviations 1 = 1_{C}, n = n_{C}, t = t_{C}.

2.1. Identities in arbitrary conic algebras. The following identities, some of which have been recorded before, are assembled here for the convenience of the reader and either hold by definition or are straightforward to check.

(1)  n_{C}(1_{C}) = 1_{C},
(2)  t_{C}(1_{C}) = 2,
(3)  t_{C}(x) = n_{C}(1_{C}, x),
(4)  x^{2} = t_{C}(x)x − n_{C}(x)1_{C},
(5)  x ∘ y := xy + yx = t_{C}(x)y + t_{C}(y)x − n_{C}(x, y)1_{C},
(6)  x^{∗} = t_{C}(x)1_{C} − x,
(7)  xx^{∗} = n_{C}(x)1_{C},
(8)  n_{C}(x^{∗}) = n_{C}(x).

2.2. Identities in flexible conic algebras. We now assume that C is flexible. By McCrimmon [33, 3.4,Thm. 3.5], this implies the following relations:

(1)  n_{C}(xy, x) = n_{C}(x)t_{C}(y) = n_{C}(yx, x),
(2)  n_{C}(x, zy^{∗}) = n_{C}(xy, z) = n_{C}(y, x^{∗}z),
(3)  n_{C}(x, y) = t_{C}(xy^{∗}) = t_{C}(x)t_{C}(y) − t_{C}(xy),
(4)  t_{C}(xy) = t_{C}(yx), t_{C}(xyz) := t_{C}((xy)z) = t_{C}(x(yz)).

Moreover, the conjugation is an algebra involution of C, so we have (xy)^{∗} = y^{∗}x^{∗} for all x, y ∈ C. Dealing with flexible conic algebras has the additional advantage that this property is preserved under the Cayley-Dickson construction [33, Thm. 6.8].

2.3. Remark. By [33, 3.4], each one of the four(!) identities in (2.2.1),(2.2.2) is actually equivalent to C being flexible.

The norm of a flexible conic algebra will in general not permit composition. But we have at least the following result.

2.4. Proposition. Let C be a flexible conic algebra over k. Then

(1)  n_{C}(xy) = n_{C}(yx),
(2) n_{C}([x, y]) = 4n_{C}(xy) − t_{C}(x)^{2}n_{C}(y) − t_{C}(y)^{2}n_{C}(x) + t_{C}(xy)t_{C}(xy^{∗})

for all x, y ∈ C.

Proof. Expanding the expression n(x ∘ y) by means of (2.1.5),(2.1.3) yields

n(x ∘ y) = t(x)^{2}n(y) + t(y)^{2}n(x) + n(x, y)^{2} − t(x)t(y)n(x, y),

where flexibility allows us to invoke (2.2.3); we obtain

n(x ∘ y) = t(x)^{2}n(y) + t(y)^{2}n(x) − t(xy)t(xy^{∗}). (3)

Now let ε = ±1. Then

n(xy + εyx) = n(xy) + εn(xy, yx) + n(yx) = n(xy) + (1 − 2ε)n(yx) + εn(x ∘ y, yx),

and combining (2.1.5) with (2.2.1)(2.2.3),(2.2.4), we conclude

n(xy + εyx) = n(xy) + (1 − 2ε)n(yx)+ ε(t(x)^{2}n(y) + t(y)^{2}n(x) − t(xy)t(xy^{∗})). (4)

Comparing (3) and (4) for ε = 1 yields (1), while (1) and (4) for ε = −1 yield (2). □

2.5. Proposition. Let C be a non-degenerate conic algebra over k.

(a) C is simple as an algebra with conjugation (cf. 1.3).

(b) If ι_{C} is an algebra involution of C, in particular, if C is flexible, then C is either simple or split quadratic étale.

Proof. (a) Let I ⊆ C be an ideal with I^{∗} = I. For x ∈ I, y ∈ C we linearize (2.1.7) and obtain n_{C}(x, y)1_{C} = xy^{∗} + yx^{∗} ∈ I. Then either I = C or n_{C}(x, y) = 0 for all x ∈ I, y ∈ C, forcing I = {0} by non-degeneracy.

(b) Assuming C is not simple, we must show it has dimension 2. Since (C, ι_{C}) is simple as an algebra with involution by (a), there exists a k-algebra A such that (C, ι_{C}) ≅ (A^{op} ⊕ A, ε) as algebras with involution, ε being the exchange involution of A^{op} ⊕ A. Then A, embedded diagonally into A^{op} ⊕ A, and

H := H(C, ι_{C}) := {x ∈ C | x = x^{∗}} ⊆ C

identify canonically as vector spaces over k. In particular, not only the dimension of H but also its codimension in C agree with the dimension of A. Applying (2.1.6), we conclude that H has dimension 1 for char(k) ≠ 2 and codimension 1 for char(k) = 2. In both cases, C must be 2-dimensional. □

2.6. Proposition. Let C be a flexible conic algebra over k whose norm is anisotropic. Then either C is non-degenerate or ∂n_{C} = 0.

Proof. Since, by 1.4, all non-zero elements of C are invertible, C is a simple algebra. On the other hand, (2.2.2) shows that I := C^{⊥} ⊆ C is an ideal. If I = {0}, then C is non-degenerate. If I = C, then ∂n_{C} = 0. □

2.7. Identities in conic alternative algebras. If C is a conic alternative algebra, then by [33, p. 97] its norm permits composition:

n_{C}(xy) = n_{C}(x)_{C}n(y). (1)

Linearizing (1), we obtain

(2)  n_{C}(x_{1}y, x_{2}y) = n_{C}(x_{1}, x_{2})n_{C}(y),
(3)  n_{C}(xy_{1}, xy_{2}) = n_{C}(x)n_{C}(y_{1}, y_{2}),
(4)  n_{C}(x_{1}y_{1}, x_{2}y_{2}) + n_{C}(x_{1}y_{2}, x_{2}y_{1}) = n_{C}(x_{1}, x_{2})n_{C}(y_{1}, y_{2}).

Moreover, by [33, Prop. 3.9] and (2.2.3),

xyx = n_{C}(x, y^{∗})x − n_{C}(x)y^{∗} = t_{C}(xy)x − n_{C}(x)y^{∗}. (5)

2.8. Theorem. Let C be a conic alternative algebra over k. Then

n_{C}([x_{1}, x_{2}, x_{3}]) = 4n_{C}(x_{1})n_{C}(x_{2})n_{C}(x_{3}) − ∑ t_{C}(x_{i})^{2}n_{C}(x_{j})n_{C}(x_{l})+ (1)

∑ t_{C}(x_{i}x_{j})t_{C}(x_{i}x_{j}^{∗})n_{C}(x_{l}) − t_{C}(x_{1}x_{2})t_{C}(x_{2}x_{3})t_{C}(x_{3}x_{1})+

t_{C}(x_{1}x_{2}x_{3})t_{C}(x_{2}x_{1}x_{3})

for all x_{1}, x_{2}, x_{3} ∈ C, where both summations on the right of (1) are taken over the cyclic permutations (ijl) of (123).

Proof. Expanding

n([x_{1}, x_{2}, x_{3}]) = n((x_{1}x_{2})x_{3}) − n((x_{1}x_{2})x_{3}, x_{1}(x_{2}x_{3})) + n(x_{1}(x_{2}x_{3}))

and applying (2.7.1), we conclude

n([x_{1}, x_{2}, x_{3}]) = 2n(x_{1})n(x_{2})n(x_{3}) − n((x_{1}x_{2})x_{3}, x_{1}(x_{2}x_{3})). (2)

Turning to the second summand on the right of (2), we obtain, by (2.7.4),

n((x_{1}x_{2})x_{3}, x_{1}(x_{2}x_{3})) = n(x_{1}x_{2}, x_{1})n(x_{3}, x_{2}x_{3}) − n((x_{1}x_{2})(x_{2}x_{3}), x_{1}x_{3}),

where applying (2.2.1) to the first summand on the right yields

n((x_{1}x_{2})x_{3}, x_{1}(x_{2}x_{3})) = t(x_{2})^{2}n(x_{3})n(x_{1}) − n((x_{1}x_{2})(x_{2}x_{3}), x_{1}x_{3}). (3)

WILD PFISTER FORMS 11

Manipulating the expression (x_{1}x_{2})(x_{2}x_{3}) by means of (2.1.5) and the Moufang identities (1.1.3), we obtain

(x_{1}x_{2})(x_{2}x_{3}) = (x_{1}x_{2}) ∘ (x_{2}x_{3}) − (x_{2}x_{3})(x_{1}x_{2})
= t(x_{1}x_{2})x_{2}x_{3} + t(x_{2}x_{3})x_{1}x_{2} − n(x_{1}x_{2}, x_{2}x_{3})1 − x_{2}(x_{3}x_{1})x_{2},

where (2.7.5),(2.2.4) yield

(4)  (x_{1}x_{2})(x_{2}x_{3}) = t(x_{1}x_{2})x_{2}x_{3} + t(x_{2}x_{3})x_{1}x_{2} − n(x_{1}x_{2}, x_{2}x_{3})1−
t(x_{1}x_{2}x_{3})x_{2} + n(x_{2})(x_{3}x_{1})^{∗}.

Here we use (2.2.2),(2.1.6) to compute

n(x_{1}x_{2}, x_{2}x_{3}) = n(x_{1}, x_{2}x_{3}x_{2}^{∗}) = t(x_{2})n(x_{1}, x_{2}x_{3}) − n(x_{1}, x_{2}x_{3}x_{2})

and (2.2.3),(2.7.5) give

n(x_{1}x_{2}, x_{2}x_{3}) = t(x_{2})n(x_{1}, x_{2}x_{3}) − t(x_{2}x_{3})n(x_{1}, x_{2}) + t(x_{3}x_{1})n(x_{2})
= t(x_{1})t(x_{2})t(x_{2}x_{3}) − t(x_{2})t(x_{1}x_{2}x_{3}) − t(x_{1})t(x_{2})t(x_{2}x_{3})+
t(x_{1}x_{2})t(x_{2}x_{3}) + t(x_{3}x_{1})n(x_{2}),

hence

n(x_{1}x_{2}, x_{2}x_{3}) = t(x_{1}x_{2})t(x_{2}x_{3}) − t(x_{2})t(x_{1}x_{2}x_{3}) + t(x_{3}x_{1})n(x_{2}).

Inserting this into (4), and (4) into the second term on the right of (3), we conclude

n((x_{1}x_{2})(x_{2}x_{3}), x_{1}x_{3}) = t(x_{1}x_{2})n(x_{2}x_{3}, x_{1}x_{3}) + t(x_{2}x_{3})n(x_{1}x_{2}, x_{1}x_{3})−
t(x_{1}x_{2})t(x_{2}x_{3})t(x_{3}x_{1}) + t(x_{2})t(x_{3}x_{1})t(x_{1}x_{2}x_{3})−
t(x_{3}x_{1})^{2}n(x_{2}) − t(x_{1}x_{2}x_{3})n(x_{2}, x_{1}x_{3})+
n((x_{3}x_{1})^{∗}, x_{1}x_{3})n(x_{2})
= t(x_{1}x_{2})t(x_{1}x_{2}^{∗})n(x_{3}) + t(x_{2}x_{3})t(x_{2}x_{3}^{∗})n(x_{1})−
t(x_{1}x_{2})t(x_{2}x_{3})t(x_{3}x_{1}) + t(x_{2})t(x_{3}x_{1})t(x_{1}x_{2}x_{3})−
t(x_{3}x_{1})^{2}n(x_{2}) − t(x_{2})t(x_{3}x_{1})t(x_{1}x_{2}x_{3})+
t(x_{1}x_{2}x_{3})t(x_{2}x_{1}x_{3}) + t(x_{3}x_{1}^{2}x_{3})n(x_{2}),

where we may use (2.1.4),(2.2.3),(2.2.2) to expand

t(x_{3}x_{1}^{2}x_{3})n(x_{2}) − t(x_{3}x_{1})^{2}n(x_{2}) = t(x_{3}^{2}x_{1}^{2})n(x_{2}) − t(x_{3}x_{1})^{2}n(x_{2})
= t([t(x_{3})x_{3} − n(x_{3})1][t(x_{1})x_{1} − n(x_{1})1])n(x_{2})−
t(x_{3}x_{1})^{2}n(x_{2})
= t(x_{3})t(x_{1})t(x_{3}x_{1})n(x_{2}) − t(x_{1})^{2}n(x_{2})n(x_{3})−
t(x_{3})^{2}n(x_{1})n(x_{2}) + 2n(x_{1})n(x_{2})n(x_{3})−
t(x_{3}x_{1})^{2}n(x_{2})
= t(x_{3}x_{1})t(x_{3}x_{1}^{∗})n(x_{2}) − t(x_{1})^{2}n(x_{2})n(x_{3})−
t(x_{3})^{2}n(x_{1})n(x_{2}) + 2n(x_{1})n(x_{2})n(x_{3}).

Inserting the resulting expression

n((x_{1}x_{2})(x_{2}x_{3}), x_{1}x_{3}) = ∑ t(x_{i}x_{j})t(x_{i}x_{j}^{∗})n(x_{l}) − t(x_{1}x_{2})t(x_{2}x_{3})t(x_{3}x_{1})+
t(x_{1}x_{2}x_{3})t(x_{2}x_{1}x_{3}) − t(x_{1})^{2}n(x_{2})n(x_{3})−
t(x_{3})^{2}n(x_{1})n(x_{2}) + 2n(x_{1})n(x_{2})n(x_{3})

into (3) and (3) into (2), the theorem follows. □

Remark. The associator of a conic alternative algebra being alternating, its norm must be totally symmetric in all three variables. Since the expression t_{C}(xy^{∗}) is symmetric in x, y ∈ C by (2.2.3), this fact is in agreement with the right-hand side of (2.8.1).

3. Interlude: Pfister bilinear and quadratic forms in characteristic 2

Working over an arbitrary field k of characteristic 2, the main purpose of this section is to collect a few results on Pfister bilinear and Pfister quadratic forms that are hardly new but, at least in their present form, apparently not in the literature. The final result, Prop. 3.12, concerns quadratic forms in arbitrary characteristic and may be amusing even for experts.

Recall the following from, for example, [21, 8.5(iv)]. If one is given an anisotropic n-Pfister bilinear form b ≅ ⟨⟨α_{1}, …, α_{n}⟩⟩, then K := k(\sqrt{α_{1}}, …, \sqrt{α_{n}}) is a field of dimension 2^{n} over k such that K^{2} ⊆ k. Conversely, given such an extension K/k, we can view K as a k-vector space endowed with a k-valued quadratic form q: x ↦ x^{2}, and one checks that q is isomorphic to the quadratic form v ↦ b(v, v). This defines a bijection between isomorphism classes of extensions K/k of dimension 2^{n} with K^{2} ⊆ k and anisotropic n-quasi-Pfister quadratic forms [15, 10.4].

3.1. Unital linear forms. We now refine this bijection. Fix an extension K/k as in the previous paragraph, and a k-linear form s: K → k such that s(1_{K}) = 1, i.e., s is a retraction of the inclusion k ↪ K. (We say that s is unital.) Define a symmetric bilinear

b_{K,s}: K × K ⟶ k via b_{K,s}(u, v) := s(uv)

for u, v ∈ K; it is the transfer s_{∗}⟨1⟩ in the notation of [15, §20.A]. Moreover, it is anisotropic (hence non-degenerate) since b_{K,s}(u, u) = u^{2} for u ∈ K.

We remark that if both s and t are unital linear forms on K, then (by the nondegeneracy of b_{K,s}) there is some u ∈ K^{×} such that b_{K,s}(u, –) = t, i.e., s(uv) = t(v) for all v ∈ K.

3.2. Pfister bilinear forms. Fix a finite extension K/k such that K^{2} ⊆ k as assumed above. We compute b_{K,s} for a particular s. Fix a 2-basis \sqrt{α_{1}}, …, \sqrt{α_{n}} of K. The monomials \sqrt{α_{1}}^{i_{1}} ⋯ \sqrt{α_{n}}^{i_{n}} with i_{j} ∈ \{0, 1\} are a k-basis for K and we define s to be 1 on 1_{k} and 0 on the other monomials. One sees immediately that b_{K,s} is isomorphic to the Pfister bilinear form ⟨⟨α_{1}, …, α_{n}⟩⟩.

In fact, the s constructed above is the general case. Suppose we are given a unital linear form s : K → k; we will construct a 2-basis so that s is as in the previous paragraph. Start with A = ∅ and repeat the following loop: If k(A) = K, then we are done. Otherwise, [K : k(A)] is at least 2, hence there is some a ≠ 0 in the intersection of the k-subspaces Ker(s) and k(A)^{⊥}, orthogonal complementation relative to b_{K,s}. As a is not in k(A), A ∪ {a} is p-free by [6, §V.13.1, Prop. 3]. We replace A by A ∪ {a} and repeat.

In this way, we have proved: If [K : k] = 2^{n}, then b_{K,s} is an anisotropic n-Pfister bilinear form. Conversely, if we are given an anisotropic symmetric bilinear form ⟨⟨α_{1}, …, α_{n}⟩⟩ over k, then \sqrt{α_{1}}, …, \sqrt{α_{n}} is a 2-basis for a purely inseparable extension K/k as in the coarser correspondence recalled at the beginning of the section.

The preceding considerations can be made more precise by looking at the category of pairs (K, s), where K/k is a finite purely inseparable field extension of exponent at most 1, s: K → k is a unital linear form, and morphisms are (automatically injective) k-homomorphisms of field extensions preserving unital linear forms. The map (K, s) ↦ b_{K,s} defines a functor from this category into the category of anisotropic Pfister bilinear forms over k where morphisms are (automatically injective) isometries of bilinear forms.

3.3. Proposition. The functor just defined is an equivalence of categories.

Proof. Given two pairs (K, s), (K′, s′) of finite purely inseparable field extensions of exponent at most 1 over k with unital linear forms, we need only show that any isometry φ: b_{K,s} → b_{K′,s′} is, in fact, a field homomorphism preserving unital linear forms. We have s′(φ(u)φ(v)) = s(uv) for all u, v ∈ K, hence φ(u)^{2} = s′(φ(u)^{2}) = s(u^{2}) = u^{2} by unitality of s, s′, which implies φ(uv)^{2} = (uv)^{2} = u^{2}v^{2} = (φ(u)φ(v))^{2} and therefore φ(uv) = φ(u)φ(v) since we are in characteristic 2. Thus φ is a k-homomorphism of fields preserving unital linear forms in view of s′(φ(u)) = s′(φ(1_{K})φ(u)) = s(1_{K}u) = s(u). □

3.4. The passage to Pfister quadratic forms. Let α ∈ k be a scalar and s: K → k a unital linear form. We define the quadratic form q_{K;α,s} to be the transfer s_{∗}(⟨⟨α⟧ ⊗ K), where, as usual, ⟨⟨α⟧ stands for the binary quadratic form given on k ⊕ kj by the matrix ( \begin{matrix}1 & 1 \\ 0 & α \end{matrix}), so

⟨⟨α⟧(β + γj) = β^{2} + βγ + αγ^{2}

for β, γ ∈ k. By the projection formula [15, p. 84], q_{K;α,s} is isomorphic to b_{K,s}⊗⟨⟨α⟧. More concretely, on the vector space direct sum K ⊕ Kj of two copies of K over k, we can define q_{K;α,s} by the formula

(1)  q_{K;α,s}(u + vj) := u^{2} + s(uv) + αv^{2}  (u, v ∈ K).

3.5. Example. For K and s as in the first paragraph of 3.2 and α_{n+1} ∈ k, the form q_{K;α_{n+1},s} is isomorphic to ⟨⟨α_{1}, …, α_{n}, α_{n+1}⟧.

3.6. Theorem. For α ∈ k and s: K → k a unital linear form, q_{K;α,s} is an (n + 1)-Pfister quadratic form over k. Conversely, every anisotropic (n + 1)-Pfister quadratic form over k is isomorphic to q_{K;α,s} for some purely inseparable field extension K/k of exponent at most 1 and degree 2^{n}, some α ∈ k and some unital linear form s: K → k.

Proof. Combine 3.2 and Example 3.5. □

The Pfister quadratic forms q_{K;α,s} of 3.4 need not be anisotropic, nor can isometries between two of them be described as easily as in the case of Pfister bilinear forms (cf. Prop. 3.3).

3.7. Proposition. Given scalars α, β ∈ k and unital linear forms s, t: K → k, the following conditions are equivalent.

(i) The Pfister quadratic forms q_{K;α,s} and q_{K;β,t} are isometric.

(ii) There exist elements u_{0}, v_{0} ∈ K such that

(1)  β = u_{0}^{2} + s(u_{0}v_{0}) + αv_{0}^{2}, t(u) = s(uv_{0})  (u ∈ K).

Proof. We identify K ⊆ K ⊕ Kj canonically through the first summand.

(i) ⟹ (ii). If q := q_{K;α,s} and q′ := q_{K;β,t} are isometric, Witt’s theorem [15, Thm. 8.3] yields a bijective k-linear isometry φ: K ⊕ Kj →^{∼} K ⊕ Kj from q′ to q (so q ∘ φ = q′) that is the identity on K. Then there are elements u_{0}, v_{0} ∈ K such that

φ(j) = u_{0} + v_{0}j, (2)

and there are k-linear maps f, g: K → K such that

(3)  φ(u + vj) = (u + f(v)) + g(v)j  (u, v ∈ K).

Combining (2), (3), we conclude

f(1_{K}) = u_{0}, g(1_{K}) = v_{0}, (4)

while evaluating q ∘ φ = q′ at u + vj ∈ K ⊕ Kj with the aid of (3) and (1) yields

t(uv) + βv^{2} = f(v)^{2} + s(ug(v)) + s(f(v)g(v)) + αg(v)^{2}

for all u, v ∈ K. Setting u = 0, v = 1_{K} and observing (4), we obtain the first equation of (1). The second one now follows by setting v = 1_{K} again but keeping u ∈ K arbitrary.

(ii) ⟹ (i). s(v_{0}) = t(1_{K}) = 1 implies v_{0} ∈ K^{×}, and a straightforward verification using (1) shows that the map

K ⊕ Kj ⟶^{∼} K ⊕ Kj, u + vj ⟼ (u + u_{0}v) + (v_{0}v)j,

is a bijective isometry from q′ to q. □

3.8. Corollary. Let s: K → k be a unital linear form. Given β ∈ k and a unital linear form t: K → k, there exists an element α ∈ k such that

q_{K;β,t} ≅ q_{K;α,s}.

Proof. By 3.1, some v_{0} ∈ K satisfies t(u) = s(uv_{0}) for all u ∈ K. Now Prop. 3.7 applies. □

3.9. The Artin-Schreier map. Let s: K → k be a unital linear form. Then

℘_{K,s}: K ⟶ k, u ⟼ ℘_{K,s}(u) := u^{2} + s(u)

is called the Artin-Schreier map of K/k relative to s. It is obviously additive and becomes the usual Artin-Schreier map (in characteristic p = 2), simply written as ℘, for K = k. Given another unital linear form t: K → k, we conclude from 3.1 that there is a unique element v_{0} ∈ K^{×} satisfying s(v_{0}) = 1 and t(u) = s(uv_{0}) for all u ∈ K. Hence

℘_{K,t}(u) = v_{0}^{2}℘_{K,s}(uv_{0}^{−1})  (u ∈ K)

since the left-hand side is equal to u^{2} + s(uv_{0}) = v_{0}^{2}[(uv_{0}^{−1})^{2} + s(uv_{0}^{−1})].

With K′ := Ker(s), we obtain the orthogonal splitting K = k1_{K} ⊥ K′ relative to b_{K,s}, so the symmetric bilinear form b′_{K,s} := b_{K,s}|_{K′×K′} up to isometry is uniquely determined by b_{K,s} ≅ ⟨1⟩ ⊥ b′_{K,s}. For α ∈ k, u′ ∈ K′ we obtain

℘_{K,s}(α1_{K} + u′) = ℘(α) + b′_{K,s}(u′, u′) = ℘(α) + u^{′2}. (1)

3.10. Corollary. Let s: K → k be a unital linear form and α, β ∈ k.

(a) q_{K;α,s} is isotropic if and only if α ∈ Im(℘_{K,s}).

(b) q_{K;α,s} and q_{K;β,s} are isometric if and only if α ≡ β mod Im(℘_{K,s}).

Proof. (a) If q := q_{K;α,s} is isotropic, (3.4.1) shows u^{2} + s(uv) + αv^{2} = 0 for some u, v ∈ K not both zero, which implies v ≠ 0 and then α = (uv^{−1})^{2} + s(uv^{−1}) ∈ Im(℘_{K,s}). Conversely, α ∈ Im(℘_{K,s}) implies α = u^{2} + s(u) for some u ∈ K, forcing q(u + j) = 0 by (3.4.1).

(b) By Prop. 3.7, the two quadratic forms are isometric iff (3.7.1) holds with t = s. But this forces v_{0} = 1_{K}, and (3.7.1) becomes equivalent to α ≡ β mod Im(℘_{K,s}). □

3.11. Remark. (a) For a symmetric bilinear form b on a vector space V over k, we adopt the usual notation from the theory of quadratic forms by writing D̃(b) := {b(v, v)|v ∈ V } [15, Def. 1.12] as an additive subgroup of k (recall char(k) = 2). With this notation, and observing (3.9.1), Cor. 3.10 (a) amounts to saying that the quadratic form q_{K;α,s} is isotropic if and only if α belongs to Im(℘) + D̃(b′_{K,s}); written in this way, Cor. 3.10 (a) agrees with [15, Lemma 9.11].

(b) The definition of the quadratic forms q_{K;α,s} in 3.4 as well as of the Artin-Schreier map in 3.9 make sense also if K has infinite degree over k, and the isomorphism q_{K;α,s} ≅ b_{K,s} ⊗ ⟨⟨α⟧ as well as Cor. 3.10 (a) continue to hold under these more general circumstances.

We close this section with an observation that will play a useful role in the discussion of types for composition algebras over 2-Henselian fields, cf. in particular 12.7 below. For the remainder of this section, we dispense ourselves from the overall restriction that k have characteristic 2.

3.12. Proposition. Let q be a Pfister quadratic form over a field k of arbitrary characteristic and suppose q_{1}, q_{2} are Pfister quadratic subforms of q with dim(q_{1}) ≤ dim(q_{2}). Then there are Pfister bilinear forms b_{1}, b_{2} over k such that

b_{2} ⊗ b_{1} ⊗ q_{1} ≅ q ≅ b_{2} ⊗ q_{2}.

In particular, if dim(q_{1}) = dim(q_{2}), then b ⊗ q_{1} ≅ q ≅ b ⊗ q_{2} for some Pfister bilinear form b over k.

Proof. We may assume that q is anisotropic. Writing q: V → k, q_{i} = q|_{V_{i}}: V_{i} → k for i = 1, 2, with a vector space V of dimension n over k and subspaces V_{i} ⊆ V of dimension n_{i} (0 ≤ n_{1} ≤ n_{2} ≤ n), we then argue by induction on r := n − n_{1}. If r = 0, we put b_{1} := b_{2} := ⟨1⟩. Now suppose r > 0. Since the q_{i} represent 1, there are e_{i} ∈ V_{i} with q(e_{i}) = q_{i}(e_{i}) = 1. Now Witt’s theorem [15, Thm. 8.3] yields an orthogonal transformation f ∈ O(V, q) with f(e_{1}) = e_{2}, and replacing q_{1} by q_{1} ∘f^{−1}|_{f(V_{1})} : f(V_{1}) → k if necessary, we may assume e_{1} = e_{2} ∈ V_{1} ∩V_{2}. Assuming n_{2} < n, and taking orthogonal complements relative to ∂q, we obtain

dim_{k}(V_{1}^{⊥} ∩ V_{2}^{⊥}) = dim_{k} ((V_{1} + V_{2})^{⊥}) = 2^{n} − dim_{k}(V_{1} + V_{2})
= 2^{n} − dim_{k}(V_{1}) − dim_{k}(V_{2}) + dim_{k}(V_{1} ∩ V_{2}) > 2^{n} − 2^{n_{1}} − 2^{n_{2}}
≥ 2^{n} − 2^{n_{2}+1} ≥ 0.

Hence we can find a non-zero vector j ∈ V_{1}^{⊥} that also belongs to V_{2}^{⊥} if n_{2} < n. Setting μ := −q(j) ∈ k^{×}, we conclude from [15, Lemma 23.1] that q′_{1} := ⟨⟨μ⟩⟩⊗q_{1} is a subform of q, and the same holds true for q′_{2} := ⟨⟨μ⟩⟩ ⊗ q_{2} unless n_{2} = n, in which case we put q′_{2} := q_{2} = q. In any event, the induction hypothesis yields Pfister bilinear forms b′_{1}, b′_{2} over k such that b′_{2} is a subform of b′_{1}, b′_{2} = ⟨1⟩ for n_{2} = n and b′_{1} ⊗ q′_{1} ≅ q ≅ b′_{2} ⊗ q′_{2}. Hence b_{1} := b′_{1} ⊗ ⟨⟨μ⟩⟩,

b_{2} := \begin{cases}⟨1⟩ & for n_{2} = n, \\ b′_{2} ⊗ ⟨⟨μ⟩⟩ & for n_{2} < n \end{cases}

are Pfister bilinear forms that satisfy b_{1} ⊗q_{1} ≅ b′_{1} ⊗⟨⟨μ⟩⟩⊗q_{1} ≅ b′_{1} ⊗q′_{1} ≅ q ≅ b_{2} ⊗q_{2}, completing the induction step since b_{2} obviously is a subform of b_{1}. □

4. A non-orthogonal Cayley-Dickson construction

By the embedding property 1.11, the Cayley-Dickson construction may be regarded as a tool to recover the structure of a composition algebra C having dimension 2^{n}, n = 1, 2, 3, from a composition subalgebra B ⊆ C of dimension 2^{n−1}: there exists a scalar μ ∈ k^{×} with C ≅ Cay(B, μ). In the present section, a similar construction will be developed achieving the same objective when B is replaced by an inseparable subfield of degree 2^{n−1}; concerning the existence of such subfields, see 1.13. The construction we are going to present is less general but more intrinsic than the one recently investigated by Pumplün [43]. Throughout this section, we work over a field k of characteristic 2.

4.1. Unital linear forms and conic algebras. Let K/k be a purely inseparable field extension of exponent at most 1, C a conic alternative k-algebra containing K as a unital subalgebra and let l ∈ C. Since K has trivial conjugation by 1.6 and C satisfies (2.2.3), the relation

s_{l}(u) := n_{C}(u, l) = t_{C}(ul) (1)

holds for all u ∈ K and defines a linear form s_{l}: K → k, which is unital in the sense of 3.1 if t_{C}(l) = 1. In this case, s_{l} is called the unital linear form on K associated with l.

The following proposition paves the way for the non-orthogonal Cayley-Dickson construction we have in mind.

4.2. Proposition. Let C be a conic alternative algebra over k and K ⊆ C a purely inseparable subfield of exponent at most 1. Suppose l ∈ C satisfies t_{C}(l) = 1, put μ := n_{C}(l) ∈ k and write s := s_{l} for the unital linear form on K associated with l. Then B := K + Kl ⊆ C is the subalgebra of C generated by K and l. More precisely, the relations

(1)  (vl)u = s(u)v + uv + (uv)l,
(2)  u(vl) = s(uv)1_{C} + s(u)v + s(v)u + uv + (uv)l,
(3)  (v_{1}l)(v_{2}l) = s(v_{1}v_{2})1_{C} + s(v_{1})v_{2} + s(v_{2})v_{1} + (1 + μ)v_{1}v_{2}+
(s(v_{1}v_{2})1_{C} + s(v_{1})v_{2} + v_{1}v_{2})l

hold for all u, v, v_{1}, v_{2} ∈ K.

Proof. The assertion about B will follow once we have established the relations (1)−(3). To do so, we first prove

(4)  u(vl) + (vl)u = s(uv)1_{C} + s(v)u.

Since K has trace zero, we combine the definition of s with (2.1.5),(2.2.2) and obtain

u(vl) + (vl)u = t_{C}(u)vl + t_{C}(vl)u − n_{C}(u, vl)1_{C}
= s(v)u − n_{C}(v^{∗}u, l)1_{C} = s(v)u + n_{C}(uv, l)1_{C}
= s(uv)1_{C} + s(v)u,

giving (4). In order to establish (1), it suffices to show

(5)  (vx)u = t_{C}(ux)v + t_{C}(x)uv + (uv)x  (u, v ∈ K, x ∈ C)

since this implies (1) for x = l. The assertion being obvious for u = v = 1, we may assume K ≠ k, forcing k to be infinite. By Zariski density, we may therefore assume that x is invertible. Then the Moufang identities (1.1.3) combine with (2.7.5), (2.1.4), the right alternative law (1.1.2) and (4.1.1) to imply

((vx)u)x = v(xux) = v(t_{C}(xu)x − n_{C}(x)u^{∗}) = t_{C}(ux)vx + n_{C}(x)uv
= t_{C}(ux)vx + (uv)(t_{C}(x)x + x^{2}) = (t_{C}(ux)v + t_{C}(x)uv + (uv)x)x,

hence (5). Combining (4) and (1), we now obtain (2). Finally, making use of the Moufang identities again and of (1),(4),(4.1.1), we conclude

(v_{1}l)(v_{2}l) = (v_{1}l + lv_{1})(v_{2}l) + (lv_{1})(v_{2}l) = (s(v_{1})1_{C} + s(1_{C})v_{1})(v_{2}l) + l(v_{1}v_{2})l
= s(v_{1})v_{2}l + v_{1}(v_{2}l) + t_{C}((v_{1}v_{2})l)l − n_{C}(l)(v_{1}v_{2})^{∗}
= s(v_{1})v_{2}l + v_{1}(v_{2}l) + s(v_{1}v_{2})l + μv_{1}v_{2}.

Applying (2) to the second summand on the right gives (3). □

4.3. Embedding inseparable field extensions into conic algebras. We now look at the converse of the situation described in Prop. 4.2. Let K/k be a purely inseparable field extension of exponent at most 1. Suppose we are given a unital linear form s : K → k and a scalar μ ∈ k. Inspired by the relations (4.2.1)−(4.2.3), we now observe the obvious fact that the vector space direct sum K ⊕ Kj of two copies of K carries a unique unital non-associative k-algebra structure

C := Cay(K; μ, s)

into which K embeds as a unital subalgebra through the first summand such that the relations

(1)  (vj)u = (s(u)v + uv) + (uv)j,
(2)  u(vj) = (s(uv)1_{K} + s(u)v + s(v)u + uv) + (uv)j,
(3)  (v_{1}j)(v_{2}j) = (s(v_{1}v_{2})1_{K} + s(v_{1})v_{2} + s(v_{2})v_{1} + (1 + μ)v_{1}v_{2})+
(s(v_{1}v_{2})1_{K} + s(v_{1})v_{2} + v_{1}v_{2})j

hold for all u, v, v_{1}, v_{2} ∈ K. Adding (1) to (2), we obtain

u ∘ (vj) = u(vj) + (vj)u = s(uv)1_{K} + s(v)u. (4)

4.4. Proposition. With the notations and assumptions of 4.3, C = Cay(K; μ, s) is a non-degenerate flexible conic k-algebra with norm, polarized norm, trace, conjugation respectively given by the formulas

(1)  n_{C}(u + vj) = u^{2} + s(uv) + μv^{2},
(2)  n_{C}(u_{1} + v_{1}j, u_{2} + v_{2}j) = s(u_{1}v_{2}) + s(u_{2}v_{1}),
(3)  t_{C}(u + vj) = s(v),
(4)  (u + vj)^{∗} = s(v)1_{K} + u + vj

for all u, u_{1}, u_{2}, v, v_{1}, v_{2} ∈ K. Moreover, n_{C} = q_{K;μ,s} (cf. 3.4), and this is an (n + 1)-Pfister quadratic form if K has finite degree 2^{n} over k.

Proof. The right-hand side of (1) defines a quadratic form n : C → k whose polarization is given by the right-hand side of (2). In particular, setting t := ∂n(1_{C},- −), we obtain t(u + vj) = s(v) and then, using (4.3.3),(4.3.4),

(u + vj)^{2} = u^{2} + u ∘ (vj) + (vj)^{2} = u^{2} + s(uv)1_{C} + s(v)u + v^{2} + (1 + μ)v^{2} + s(v)vj
= s(v)(u + vj) + (u^{2} + s(uv) + μv^{2})1_{C}
= t(u + vj)(u + vj) − n(u + vj)1_{C}.

Thus C is indeed a conic k-algebra, and (1)−(4) hold. Since K is a field and s is unital, forcing b_{K,s} to be non-degenerate by 3.1, we conclude from (2) that ∂n_{C} is non-degenerate as well. The final statement of the proposition follows from comparing (1) with (3.4.1) and applying Thm. 3.6. It therefore remains to show that C is flexible. We do so by invoking Remark 2.3 and verifying the first relation of (2.2.1). Letting u, v, w ∈ K be arbitrary and setting x = u + vj, we may assume y = w or y = wj. Leaving the former case to the reader, we apply (1),(3),(4.3.1−3) and compute

n_{C}(xy, x) − n_{C}(x)t_{C}(y) = n_{C}(u(wj) + (vj)(wj), u + vj) − n_{C}(u + vj)t_{C}(wj)
= n_{C}(s(uw)1_{K} + s(u)w + s(w)u + uw + s(vw)1_{K} + s(v)w + s(w)v

+ (1 + μ)vw + (uw + s(vw)1_{K} + s(v)w + vw)j, u + vj) + (u^{2} + s(uv) + μv^{2})s(w)

= u^{2}s(w) + s(u)s(vw) + s(v)s(uw) + s(uvw) + s(v)s(uw) + s(u)s(vw)
+ s(w)s(uv) + s(uvw) + s(v)s(vw) + s(v)s(vw) + v^{2}s(w) + v^{2}s(w)
+ μv^{2}s(w) + u^{2}s(w) + s(w)s(uv) + μv^{2}s(w)
= 0,

which completes the proof. □

Prop. 4.2 not only serves to illuminate the intuitive background of the nonorthogonal Cayley-Dickson construction presented in 4.3 and Prop. 4.4. It also allows for the following application.

4.5. Proposition. Let C be a conic alternative algebra over k and K ⊆ C a purely inseparable subfield of exponent at most 1. Suppose l ∈ C satisfies t_{C}(l) = 1, put μ := n_{C}(l) ∈ k and write s := s_{l} for the unital linear form on K associated with l. Then there is a unique homomorphism

φ : Cay(K; μ, s) ⟶ C

extending the identity of K and satisfying φ(j) = l. Moreover, φ is injective, and its image is the subalgebra of C generated by K and l.

Proof. The uniqueness assertion being obvious, define φ : Cay(K; μ, δ) → C by φ(u + vj) := u + vl for u, v ∈ K. Then φ is a k-linear map whose image by Prop. 4.2 agrees with the subalgebra of C generated by K and l. But φ is also a homomorphism of algebras, which follows by comparing (4.3.1)−(4.3.3) with the corresponding relations (4.2.1)−(4.2.3). It remains to show that φ is injective, i.e., that u + vl = 0 for u, v ∈ K implies u = v = 0. Otherwise, we would have v ≠ 0, which implies 0 = v^{−1}(u + vl) = v^{−1}u + l and applying t_{C} yields a contradiction. □

It is a natural question to ask for conditions that are necessary and sufficient for a non-orthogonal Cayley-Dickson construction as in 4.3 to be a composition algebra. The answer is given by the following result.

4.6. Theorem. Let K/k be a purely inseparable field extension of exponent at most 1, s: K → k a unital linear form and μ ∈ k a scalar. We put C := Cay(K; μ, s).

(a) The map f_{C}: K^{4} → k defined by

f_{C}(u_{1}, u_{2}, u_{3}, u_{4}) := s(u_{1}u_{2}u_{3}u_{4}) + ∑ s(u_{i})s(u_{j}u_{l}u_{m}) + ∑ s(u_{i}u_{j})s(u_{l}u_{4})

for u_{1}, u_{2}, u_{3}, u_{4} ∈ K, where the first (resp. second) sum on the right is taken over all cyclic permutations ijlm (resp. ijl) of 1234 (resp. 123), is an alternating quadri-linear map. Setting K̇ := K/k1_{K}, f_{C} induces canonically an alternating quadri-linear map ḟ_{C}: K̇^{4} → k. Moreover, the relation

n_{C}(xy) = n_{C}(x)n_{C}(y) + f_{C}(u_{1}, u_{2}, v_{1}, v_{2}) (1)

holds for all x = u_{1} + v_{1}j, y = u_{2} + v_{2}j ∈ C with u_{i}, v_{i} ∈ K, i = 1, 2.

(b) The following conditions are equivalent.

(i) C is a composition algebra.

(ii) f_{C} = 0. (iii) [K : k] ≤ 4.

Proof. (a) Setting f := f_{C}, which is obviously quadri-linear, it is straightforward to check that it is alternating as well (hence symmetric since we are in characteristic 2) and satisfies f(1_{K}, u_{2}, u_{3}, u_{4}) = 0 for all u_{i} ∈ K, i = 1, 2, 3. This proves existence of ḟ with the desired properties. It remains to establish (1). Subtracting the first summand on the right from the left, and expanding the resulting expression in the obvious way, we conclude that it decomposes into the sum of terms

n_{C}((vj)u)− n_{C}(vj)n_{C}(u),
n_{C}(u(vj))− n_{C}(u)n_{C}(vj),
n_{C}((v_{1}j)(v_{2}j))− n_{C}(v_{1}j)n_{C}(v_{2}j),
n_{C}(u_{1}u_{2}, u_{1}(v_{2}j))− n_{C}(u_{1})n_{C}(u_{2}, v_{2}j),
n_{C}(u_{1}u_{2}, (v_{1}j)u_{2}))− n_{C}(u_{1}, v_{1}j)n_{C}(u_{2}),
n_{C}(u_{1}(v_{2}j), (v_{1}j)(v_{2}j))− n_{C}(u_{1}, v_{1}j)n_{C}(v_{2}j),
n_{C}((v_{1}j)u_{2}, (v_{1}j)(v_{2}j))− n_{C}(v_{1}j)n_{C}(u_{2}, v_{2}j),
n_{C}(u_{1}u_{2}, (v_{1}j)(v_{2}j)) + n_{C}(u_{1}(v_{2}j), (v_{1}j)u_{2})− n_{C}(u_{1}, v_{1}j)n_{C}(u_{2}, v_{2}j).

A tedious but routine computation, involving (4.2.1)−(4.2.3) and (4.4.1),(4.4.2) shows that all but the very last one of these expressions are equal to zero. Hence (1) follows from

n_{C}(u_{1}u_{2}, (v_{1}j)(v_{2}j)) + n_{C}(u_{1}(v_{2}j), (v_{1}j)u_{2})
= n_{C}(u_{1}u_{2}, (s(v_{1}v_{2})1_{K} + s(v_{1})v_{2} + s(v_{2})v_{1} + (1 + μ)v_{1}v_{2})+

(s(v_{1}v_{2})1_{K} + s(v_{1})v_{2} + v_{1}v_{2})j)+

n_{C}((s(u_{1}v_{2})1_{K} + s(u_{1})v_{2} + s(v_{2})u_{1} + u_{1}v_{2}) + (u_{1}v_{2})j, (s(u_{2})v_{1} + u_{2}v_{1}) + (u_{2}v_{1})j)

= s(u_{1}u_{2})s(v_{1}v_{2}) + s(v_{1})s(u_{1}u_{2}v_{2}) + s(u_{1}u_{2}v_{1}v_{2}) + s(u_{1}v_{2})s(u_{2}v_{1})+

s(u_{1})s(u_{2}v_{1}v_{2}) + s(v_{2})s(u_{1}u_{2}v_{1}) + s(u_{1}u_{2}v_{1}v_{2}) + s(u_{2})s(u_{1}v_{1}v_{2}) + s(u_{1}u_{2}v_{1}v_{2})

= s(u_{1}u_{2}v_{1}v_{2}) + s(u_{1})s(u_{2}v_{1}v_{2}) + s(u_{2})s(v_{1}v_{2}u_{1}) + s(v_{1})s(v_{2}u_{1}u_{2}) + s(v_{2})s(u_{1}u_{2}v_{1})+

s(u_{1}u_{2})s(v_{1}v_{2}) + s(u_{2}v_{1})s(u_{1}v_{2}) + s(v_{1}u_{1})s(u_{2}v_{2}) + s(u_{1}v_{1})s(u_{2}v_{2})

= f(u_{1}, u_{2}, v_{1}, v_{2}) + n_{C}(u_{1}, v_{1}j)n_{C}(u_{2}, v_{2}j).

(b) (i) ⟹ (iii) follows from the fact that composition algebras have dimension at most 8 and that the dimension of C is twice the degree of K/k. (iii) ⟹ (ii) follows from the fact that K̇ has dimension at most 3, so any alternating quadri-linear linear map on K̇ must be zero. (ii) ⟹ (i) follows immediately from (1) since ∂n_{C} by Prop. 4.4 is non-degenerate. □

Remark. Thm. 4.6 (b) can be proved without recourse to any a priori knowledge of composition algebras. To do so, it suffices to show directly that [K : k] > 4 implies f_{C} ≠ 0, which can be done quite easily. We omit the details.

5. The Skolem-Noether theorem for inseparable subfields

The non-orthogonal Cayley-Dickson construction introduced in the preceding section will be applied in two ways. Recalling from [47] that every isomorphism between composition subalgebras of a composition algebra C can be extended to an automorphism of C, the aim of our first application will be to derive an analogous result where the composition subalgebras are replaced by inseparable subfields. We begin by exploiting more fully our description of Pfister quadratic forms presented in Section 3 within the framework of composition algebras.

Throughout we continue to work over an arbitrary field k of characteristic 2.

5.1. Comparing non-orthogonal Cayley-Dickson constructions. Let K/k be a purely inseparable field extension of exponent at most 1 and degree 2^{n−1}, 1 ≤ n ≤ 3. Suppose we are given scalars μ, μ′ ∈ k and unital linear forms s, s′: K → k. By Thm. 4.6, C := Cay(K; μ, s) and C′ := Cay(K; μ′, s′) are composition algebras over k; the notational conventions of 4.3 will remain in force for C and will be extended to C′ = K ⊕ Kj′ in the obvious manner. By a K-isomorphism from C to C′ we mean an isomorphism that induces the identity on K; we say C and C′ are K-isomorphic if a K-isomorphism from C to C′ exists.

5.2. Proposition. With the notations and assumptions of 5.1, we have:

(a) C is split if and only if μ ∈ Im(℘_{K,s}).

(b) C and C′ are K-isomorphic if and only if there exist u_{0}, v_{0} ∈ K such that

(1)  μ′ = u_{0}^{2} + s(u_{0}v_{0}) + μv_{0}^{2}, s′(u) = s(uv_{0})  (u ∈ K).

Proof. (a) C is split iff n_{C} is isotropic iff μ ∈ Im(℘_{K,s}) by Cor. 3.10 (a) and Prop. 4.4.

(b) If C and C′ are K-isomorphic, their norms are isometric, so by Prop. 3.7, some u_{0}, v_{0} ∈ K satisfy (1). Conversely, let this be so. Setting l := u_{0} + v_{0}j ∈ C and combining (4.4.1)−(4.4.3)with (1), we conclude t_{C}(l) = s(v_{0}) = s′(1_{K}) = 1, n_{C}(l) = u_{0}^{2} + s(u_{0}v_{0}) + μv_{0}^{2} = μ′ and n_{C}(u, l) = s(uv_{0}) = s′(u) for all u ∈ K, so Prop. 4.5 yields a unique K-isomorphism C′ →^{∼} C sending j′ to l. □

5.3. Remark. While the set-up described in 5.1 above extends to the case of allowing purely inseparable extensions of exponent 1 and arbitrary degree 2^{n−1}, n ≥ 1, in the obvious manner (replacing composition algebras by flexible conic ones in the process), our methods of proof become unsustainable in this generality. A typical example is provided by the proof of Prop. 5.2 (b), where a K-isomorphism from C′ to C sending j′ to l in general does not exist unless n ≤ 3. Indeed, assuming that every element l ∈ C of trace 1 allows a K-isomorphism

φ: Cay(K; n_{C}(l), ∂n_{C}(−, l)) ⟶^{∼} C

with φ(j′) = l, one computes the expression φ((vj′)u) − φ(vj′)φ(u) = 0 and arrives at the conclusion that the trilinear map defined by

(1) H(u_{1}, u_{2}, u_{3}) := s(u_{1}u_{2}u_{3})1_{K} + ∑ (s(u_{i}u_{j})u_{l} + s(u_{i})u_{j}u_{l}) + u_{1}u_{2}u_{3}

for u_{1}, u_{2}, u_{3} ∈ K is zero, the sum on the right being extended over all cyclic permutations ijl of 123. Since H is obviously alternating and satisfies the relations H(1_{K}, u_{2}, u_{3}) = H(u_{1}, u_{2}, u_{1}u_{2}) = 0 for all u_{1}, u_{2}, u_{3} ∈ K, it is easy to check (see the corresponding argument in the proof of Thm. 4.6) that H vanishes for n ≤ 3 (as it should). But for n > 3, we may choose u_{1}, u_{2}, u_{3} ∈ K to be 2-independent over k [6, V §13.2, Thm. 2], forcing the right-hand side of (1) to be a k-linear combination of linear independent vectors over k [6, V §13.2, Prop. 1], whence H(u_{1}, u_{2}, u_{3}) ≠ 0.

5.4. Proposition. Let K/k be a purely inseparable field extension of exponent at most 1 and degree 2^{n−1}, 1 ≤ n ≤ 3. Furthermore, let s: K → k be a unital linear form.

(a) If C is a composition algebra of dimension 2^{n} over k containing K as a unital subalgebra, then there exists a scalar μ ∈ k such that C and Cay(K; μ, s) are Kisomorphic.

(b) For μ, μ′ ∈ k, the following conditions are equivalent.

(i) Cay(K; μ, s) and Cay(K; μ′, s) are K-isomorphic.

(ii) Cay(K; μ, s) ≅ Cay(K; μ′, s).

(iii) μ ≡ μ′ mod Im(℘_{K,s}).

Proof. (a) Pick any l ∈ C of trace 1. Then C = K ⊕ Kl, and Prop. 4.5 yields a K-isomorphism C →^{∼} Cay(K; μ′, s′), for some μ′ ∈ k and some unital linear form s′: K → k. Hence there is a unique v_{0} ∈ K^{×} such that s′(u) = s(uv_{0}) for all u ∈ K. Setting μ := v_{0}^{−2}μ′ + ℘_{K,s}(u_{0}v_{0}^{−1}) and consulting Prop. 5.2 (b), we obtain a K-isomorphism Cay(K; μ′, s′) →^{∼} Cay(K; μ, s).

(b) While (i) ⇒ (ii) is obvious, (ii) ⇒ (iii) follows from Cor. 3.10 (b). It remains to show (iii) ⇒ (i). But (iii) implies μ′ = μ + ℘_{K,s}(u_{0}) for some u_{0} ∈ K, so (5.2.1) holds with v_{0} = 1_{K}. This gives (i). □

Every element of an octonion algebra over a field is contained in a suitable quaternion subalgebra [47, Prop. 1.6.4]. However, it doesn’t seem entirely obvious that, if the octonion algebra is split, the quaternion subalgebra can be chosen to be split as well. But, in fact, it can:

5.5. Proposition. Let C be a split octonion algebra over an arbitrary field F. Then every element of C is contained in a split quaternion subalgebra of C.

Proof. Let x ∈ C. We may assume that R := k[x] has dimension 2 over k. There are three cases [5, III §2 Prop. 3].

Case 1. R is the algebra of dual numbers. In particular, it contains zero divisors. Hence so does every quaternion subalgebra of C containing x, which therefore must be split.

Case 2. R is (quadratic) étale. Since C up to isomorphism is uniquely determined by splitness, it may be obtained from R by the Cayley-Dickson process as C ≅ Cay(R; 1, 1), which implies that x is contained in the split quaternion subalgebra Cay(R, 1) of C.

Case 3. We are left with the most delicate possibility that F = k has characteristic 2 and K := R is a purely inseparable field extension of k having exponent at most 1. Let c ∈ C be an idempotent different from 0, 1_{C}, which exists since C is split, and write B for the subalgebra of C generated by K and c. Since c has trace 1, Prop. 4.5 yields a scalar μ ∈ k, a unital linear form s: K → k and an isomorphism from B′ := Cay(K; μ, s) onto B. But B′ is a quaternion algebra by Thm. 4.6 whereas B, containing c, has zero divisors. Hence B is a split quaternion subalgebra of C containing x. □

And finally, we need a purely technical result.

5.6. Lemma. Let C be an octonion division algebra over k and suppose φ: K_{1} →^{∼} K_{2} is an isomorphism of inseparable quadratic subfields K_{1}, K_{2} ⊆ C. Then there exist inseparable subfields L_{1}, L_{2} ⊆ C of degree 4 over k such K_{i} ⊆ L_{i} for i = 1, 2 and φ extends to an isomorphism ψ: L_{1} →^{∼} L_{2}.

Proof. Given any elements y ∈ C, x_{i} ∈ K_{i} \ k1_{C} for i = 1, 2, denote by L_{i} the subalgebra of C generated by K_{i} and y. Since C has no zero divisors, L_{i} is either a composition algebra or an inseparable field extension, the latter possibility being equivalent to the trace of C vanishing identically on L_{i}. In any event, the dimension of L_{i} is either 2 or 4. Moreover, L_{i} is spanned by 1_{C}, x_{i}, y, x_{i}y as a vector space over k. Summing up, L_{i}/k is therefore an inseparable field extension of degree 4 if and only if y ∉ K_{i} satisfies the condition t_{C}(y) = t_{C}(x_{i}y) = 0. To choose y appropriately, we now write C^{0} for the space of trace zero elements in C and consider the hyperplane intersection V := C^{0} ∩ x_{1}C^{0} ∩ x_{2}C^{0} ⊆ C, which is a subspace of dimension at least 5. Since a group cannot be the union of two proper subgroups, we conclude V ⊈ K_{1} ∪ K_{2} and, accordingly, pick a y ∈ V that neither belongs to K_{1} nor to K_{2}. Then y has trace zero and y = x_{i}z_{i} for some z_{i} ∈ C^{0}, so x_{i}y = x_{i}^{2}z_{i} = n_{C}(x_{i})z_{i} ∈ kz_{i} has trace zero as well, and by the above, L_{i}/k is an inseparable field extension of degree 4 containing K_{i}. Moreover, K_{1} ≅ K_{2} implies K_{1}^{2} = K_{2}^{2}, hence

L_{1}^{2} = K_{1}^{2} + K_{1}^{2}y^{2} = K_{2}^{2} + K_{2}^{2}y^{2} = L_{2}^{2},

so there is an isomorphism ψ : L_{1} →^{∼} L_{2}, which necessarily extends φ since the only k-embedding K_{1} → L_{2} is the one induced by φ. □

After these preparations, we can now establish the Skolem-Noether theorem for inseparable subfields of composition algebras.

5.7. Theorem. (Skolem-Noether) Let C be a composition algebra over k. Then every isomorphism between inseparable subfields of C can be extended to an automorphism of C.

Proof. Write dim_{k}(C) = 2^{n}, 0 ≤ n ≤ 3 and let φ: K_{1} →^{∼} K_{2} be an isomorphism of inseparable subfields K_{1}, K_{2} ⊆ C having degree 2^{n′}, 0 ≤ n′ < n, over k. We may assume n′ > 0 and first reduce to the case n′ = n − 1. To do so, suppose the theorem holds for n′ = n − 1 and let n′ < n − 1. Then n′ = 1, n = 3, so C is an octonion algebra containing K_{1}, K_{2} as inseparable quadratic subfields. If C is split, there are split quaternion subalgebras B_{i} ⊆ C containing K_{i} for i = 1, 2 (Prop. 5.5). But B_{1}, B_{2} are isomorphic, hence conjugate under the automorphism group of C, by the classical Skolem-Noether theorem for composition algebras. Hence up to conjugation by automorphisms of C, we may assume B_{1} = B_{2} =: B. But then φ extends to an automorphism of B, which in turn extends to an automorphism of C. We are left with the case that C is a division algebra. By Lemma 5.6, there are inseparable subfields K_{i} ⊆ L_{i} ⊆ C of degree 4 (i = 1, 2) such that φ extends to an isomorphism ψ: L_{1} →^{∼} L_{2}. But ψ in turn extends to an automorphism of C, completing the reduction to the case n′ = n − 1. From now on we assume n′ = n − 1 and fix a unital linear form s_{2} : K_{2} → k. Then s_{1} := s_{2} ∘ φ: K_{1} → k is a unital linear form as well. For i = 1, 2, Prop. 5.4 (a) yields scalars μ_{i} ∈ k and K_{i}-isomorphisms ψ_{i}: Cay(K_{i}; μ_{i}, s_{i}) →^{∼} C. Now observe that the non-orthogonal Cayley-Dickson construction of 4.3 is functorial in the parameters involved. Hence φ determines canonically an isomorphism

ψ := Cay(φ): Cay(K_{1}; μ_{1}, s_{1}) ⟶^{∼} Cay(K_{2}; μ_{1}, s_{1} ∘ φ^{−1}) = Cay(K_{2}; μ_{1}, s_{2}).

Putting things together, we thus obtain an isomorphism

ψ_{2}^{−1} ∘ ψ_{1} ∘ ψ^{−1} : Cay(K_{2}; μ_{1}, s_{2}) ⟶^{∼} Cay(K_{2}; μ_{2}, s_{2}).

Applying Prop. 5.4 (b), we see that there exists a K_{2}-isomorphism

χ: Cay(K_{2}; μ_{2}, s_{2}) →^{∼} Cay(K_{2}; μ_{1}, s_{2}),

giving rise to the automorphism ϕ := ψ_{2} ∘χ^{−1} ∘ψ ∘ψ_{1}^{−1} of C, which extends φ since ψ_{2} = χ^{−1} = 1 on K_{2} and ψ = φ, ψ_{1}^{−1} = 1 on K_{1}. □

Remark. Let C be an octonion algebra over k and K ⊆ C an inseparable subfield of degree 4. Changing scalars to the algebraic closure, k̄, of k, K ⊗_{k} k̄ becomes a unital k̄-subalgebra of C̄ := C ⊗_{k} k̄ containing a 3-dimensional subalgebra N that consists entirely of nilpotent elements. Hence N ⊆ C̄ is a Borel subalgebra in the sense of [38]. The fact that all Borel subalgebras of C̄ are conjugate under its automorphism group [38, § 2, 1.] corresponds nicely with the Skolem-Noether theorem.

It is a natural question to ask how our non-orthogonal Cayley-Dickson construction can be converted into the classical orthogonal one. When dealing within the framework of composition (and not of arbitrary conic) algebras, here is a simple answer.

5.8. Orthogonalizing the non-orthogonal Cayley-Dickson construction. Let K/k be a purely inseparable field extension of exponent at most 1 and degree 2^{n}, n = 1, 2. Suppose we are given an intermediate subfield k ⊆ K′ ⊆ K of degree 2^{n−1}, a scalar μ ∈ k and a unital linear form s: K → k. Then s′ := s|_{K′}: K′ → k is a unital linear form on K′ and B := Cay(K′; μ, s′) is a composition subalgebra of C := Cay(K; μ, s). Moreover, (4.4.2) implies

B^{⊥} = K^{′⊥} ⊕ K^{′⊥}j, (1)

orthogonal complementation in K (resp. C) being taken relative to b_{K,s} (resp. ∂n_{C}). From (1) we conclude

C = Cay(B, −n_{C}(u)) = Cay(B; u^{2}) (2)

for any non-zero element u ∈ K^{′⊥}.

To be more specific, let C be an octonion algebra over k and K ⊆ C an inseparable subfield of degree 4. Pick a 2-basis a = (a_{1}, a_{2}) of K/k. By Prop. 5.4 (a), there exists a scalar μ ∈ k with C ≅ Cay(K; μ, s_{a}). On the other hand,

L = Cay(k; μ, 1_{k}) = k[t]/(t^{2} + t + μ)

is a quadratic étale k-algebra, and (2) yields

C = Cay(L; a_{1}^{2}, a_{2}^{2})

as an ordinary Cayley-Dickson process starting from L.

6. Conic division algebras in characteristic 2.

We now turn to a second application of the non-orthogonal Cayley-Dickson construction which consists in finding new examples of conic division algebras. A few comments on the historical context seem to be in order.

6.1. Conic division algebras over arbitrary fields. Among all conic division algebras, it is the composition division algebras that are particularly well understood and particularly easy to construct: by 1.9, it suffices to ensure that their norms be anisotropic. Of course, composition division algebras exist only in dimensions 1, 2, 4, 8, as do all non-associative division algebras over the reals, by the Bott-Kervaire-Milnor theorem [13, Kap. 10,§ 2]; in particular, the Cayley-Dickson process 1.12 leads to conic algebras

Cay(ℝ; μ_{1}, …, μ_{n}), μ_{1} = ⋯ = μ_{n} = −1  (n ∈ ℤ, n ≥ 1)

over the reals whose norms are positive definite (hence anisotropic) but which fail to be division algebras unless n ≤ 3. Hence it is natural to ask for examples of conic division algebras in dimensions other than 1, 2, 4, 8, over fields other than the reals.

From the point of view of non-associative algebras, conic division algebras that are not central, like purely inseparable field extensions of characteristic 2 and exponent 1 as in 1.6, are not particularly interesting. Over appropriate fields of characteristic not 2, the first examples of central conic division algebras in all dimensions 2^{n}, n = 0, 1, 2 … are apparently due to Brown [7, pp. 421-422]. They all arise from the base field, an iterated Laurent series field in finitely many variables, by the Cayley-Dickson process; generalizations of these examples will be discussed in Example 10.7 below. Other examples of central conic division algebras in dimension 16, using a refinement of the Cayley-Dickson construction, have been exhibited by Becker [3, Satz 16]. Examples of central commutative conic division algebras in characteristic 2 and all dimensions 2^{n}, n ≥ 0, have been constructed by Albert [2, Thm. 2]. These algebras are closely related to purely inseparable field extensions of exponent 1 since their norms bilinearize to zero and hence degenerate when extending scalars to the algebraic closure.

In view of the preceding results one may ask whether the dimension of a finitedimensional conic division algebra is always a power of 2. Though a feeble result along these lines has been obtained by Petersson [39], the answer to this question doesn’t seem to be known.

In this paper, two classes of conic division algebras in all dimensions 2^{n}, n = 0, 1, 2 …, will be constructed. The examples of the first class, to be discussed in the present section, depend strongly on the non-orthogonal Cayley-Dickson construction, hence exist only in characteristic 2 but differ from Albert’s by being central and highly non-commutative and by allowing arbitrary anisotropic Pfister quadratic forms as their norms, which in particular remain non-singular under all scalar extensions. The second class of examples will be discussed in 10.7 and 10.15 below.

6.2. Notations and conventions. For the remainder of this section, we fix a base field k of characteristic 2, a purely inseparable field extension K/k of exponent at most 1, a scalar μ ∈ k and a unital linear form s: K → k to consider the non-orthogonal Cayley-Dickson construction C := Cay(K; μ, s) as in 4.3. We put [K : k] = 2^{n}, n = 0, 1, 2, … and explicitly allow the possibility n = ∞, i.e., that K has infinite degree over k. The following proposition paves the way for the application we have in mind.

6.3. Proposition. With the notations and conventions of 6.2, the following assertions hold.

(a) C is locally finite-dimensional.

(b) For n ≥ 2, every element of C belongs to an octonion subalgebra of C.

(c) C is central simple for n ≥ 1 and has trivial nucleus for n ≥ 2.

Proof. (a) It suffices to note that finitely many elements x_{i} = u_{i} + v_{i}j ∈ C with u_{i}, v_{i} ∈ K (1 ≤ i ≤ m) are contained in Cay(K′; μ, s|_{K′}) (K′ := k(u_{1}, v_{1}, …, u_{m}, v_{m})), which is a subalgebra of C having dimension at most 2^{2m+1}.

(b) Let x = u + vj ∈ C, u, v ∈ K. Then there is a subfield K′ ⊆ K containing u, v and having degree 4 over k, so C′ := Cay(K′; μ, s|_{K′}) ⊆ C by Thm. 4.6 is an octonion subalgebra containing x.

(c) Standard properties of composition algebras allow us to assume n > 2. Combining Props. 2.5, 4.4 we see that C is simple. Let x ∈ Nuc(C) and apply (b) to pick an octonion subalgebra C′ ⊆ C containing x. Since C′ has trivial nucleus [47, Prop. 1.9.2], we conclude x ∈ k1_{C}, so C has trivial nucleus as well; in particular, C is central. □

Referring the reader to our version of the Artin-Schreier map (3.9, Remark 3.11), we can now state the main result of this section.

6.4. Theorem. With the notations and conventions of 6.2, the following conditions are equivalent.

(i) C is a division algebra.

(ii) n_{C} is anisotropic. (iii) μ ∉ Im(℘_{K,s}).

Proof. The implication (i) ⇒ (ii) follows from 1.5, while (ii) and (iii) are equivalent by Cor. 3.10 (a) and Remark 3.11 (b). It remains to prove (ii) ⟹ (i). Since C is locally finite-dimensional by Prop. 6.3 (a), it suffices to show that there are no zero divisors, so suppose x_{1}, x_{2} ∈ C satisfy x_{1}x_{2} = 0. By (ii) and (2.4.1), this implies x_{2}x_{1} = 0, and from (2.1.5) we conclude 0 = x_{1}x_{2} + x_{2}x_{1} = t_{C}(x_{1})x_{2} + t_{C}(x_{2})x_{1} − n_{C}(x_{1}, x_{2})1_{C}. If t_{C}(x_{1}) ≠ 0, this yields x_{2} ∈ k[x_{1}], hence x_{2} = 0 since n_{C} being anisotropic implies that k[x_{1}] is a field. By symmetry, we may therefore assume t_{C}(x_{1}) = t_{C}(x_{2}) = n_{C}(x_{1}, x_{2}) = 0. Write x_{i} = u_{i} + v_{i}j with u_{i}, v_{i} ∈ K for i = 1, 2. Then s(v_{i}) = 0 by (4.4.3), s(u_{1}v_{2}) = s(u_{2}v_{1}) by (4.4.2), and if v_{1} = 0 or v_{2} = 0, Thm. 4.6 (a) yields n_{C}(x_{1})n_{C}(x_{2}) = n_{C}(x_{1}x_{2}) = 0, hence x_{1} = 0 or x_{2} = 0. We are thus reduced to the case

v_{1} ≠ 0 ≠ v_{2}, s(v_{1}) = s(v_{2}) = 0, s(u_{1}v_{2}) = s(u_{2}v_{1}). (1)

Next we use (4.3.1−3) and (1) to expand (u_{1} + v_{1}j)(u_{2} + v_{2}j) = 0. A short computation gives

u_{1}u_{2} + s(u_{1}v_{2})1_{K} + s(u_{1})v_{2} + u_{1}v_{2} + s(u_{2})v_{1} + u_{2}v_{1} + s(v_{1}v_{2})1_{K} + (1 + μ)v_{1}v_{2} = 0,
(2)  u_{1}v_{2} + u_{2}v_{1} + s(v_{1}v_{2})1_{K} + v_{1}v_{2} = 0.

Adding these two relations, we obtain

u_{1}u_{2} + s(u_{1}v_{2})1_{K} + s(u_{1})v_{2} + s(u_{2})v_{1} + μv_{1}v_{2} = 0 (3)

and note that (2),(3) are symmetric in the indices 1, 2 by (1). We now claim: (∗) The subfield K′ of K/k generated by u_{1}, u_{2}, v_{1}, v_{2} is spanned by

1_{K}, u_{1}, u_{2}, v_{1}, v_{2}, u_{1}v_{2}, u_{2}v_{1} (4)

as a vector space over k Suppose for the time being that this claim has been proved. Then the field extension K′/k has degree at most 7. Being purely inseparable at the same time, it has, in fact, degree at most 4. By Thm. 4.6 (b), C′ := Cay(K′; μ, s|_{K′}) ⊆ C is therefore a composition subalgebra containing x_{1}, x_{2} and inheriting its anisotropic norm from C. Thus C′ is a division algebra, and x_{1}x_{2} = 0 implies x_{1} = 0 or x_{2} = 0, as desired.

We are thus reduced to showing (∗). Writing V for the linear span of the vectors in (4), it suffices to show that V ⊆ K is a k-subalgebra, i.e., that the product of any two distinct elements in (4) belongs to V. Since v_{1}v_{2} ∈ V by (2), hence u_{1}u_{2} ∈ V by (3), this will follow once we have shown that

v_{p}^{2}u_{q}v_{q}, u_{p}u_{q}v_{p}, u_{p}v_{p}v_{q}  ({p, q} = {1, 2})

all belong to V. By symmetry, we may assume p = 1, q = 2. Multiplying (2) by u_{2}v_{1}, we obtain

u_{1}u_{2}v_{1}v_{2} + u_{2}^{2}v_{1}^{2} + s(v_{1}v_{2})u_{2}v_{1} + v_{1}^{2}u_{2}v_{2} = 0,

forcing v_{1}^{2}u_{2}v_{2} ≡ u_{1}u_{2}v_{1}v_{2} mod V. But multiplying (3) by v_{1}v_{2} implies u_{1}u_{2}v_{1}v_{2} ∈ V, so we have v_{1}^{2}u_{2}v_{2} ∈ V as well. Moreover, multiplying (2) first by u_{1}, then by v_{1}, yields

u_{1}u_{2}v_{1} = u_{1}^{2}v_{2} + s(v_{1}v_{2})u_{1} + u_{1}v_{1}v_{2} ≡ v_{1}^{2}u_{2} + s(v_{1}v_{2})v_{1} + v_{1}^{2}v_{2} ≡ 0 mod V.

Hence also u_{1}v_{1}v_{2} = u_{1}^{2}v_{2} + u_{1}u_{2}v_{1} + s(v_{1}v_{2})u_{1} ∈ V, which completes the proof. □

6.5. Corollary. For a Pfister quadratic form q over a field of characteristic 2 to be the norm of a conic division algebra it is necessary and sufficient that q be anisotropic. □

6.6. Examples. Letting k be the field of rational functions in countably many variables t_{1}, t_{2}, t_{3}, … over any field of characteristic 2, e.g., over 𝔽_{2}, the (n + 1)- Pfister quadratic forms ⟨⟨t_{1}, …, t_{n}, t_{n+1}⟧, n ≥ 0, by standard arguments are easily seen to be anisotropic, and we obtain central flexible conic division algebras over k in all dimensions 2^{n}, n = 0, 1, 2, . . ..

Part II. 2-Henselian base fields

7. Pointed quadratic spaces over 2-Henselian fields.

In this section, we recast the conceptual foundations for the study of quadratic forms over Henselian fields in the setting of pointed quadratic spaces. Our subsequent considerations also fit naturally into the valuation theory of Jordan division rings [40] when specialized to the Jordan algebras of pointed quadratic spaces over Henselian fields.

7.1. Round quadratic forms. For the time being, we work over a field k that is completely arbitrary. We recall from [15, § 9A, p. 52] that a finite-dimensional quadratic form q over k is said to be round if all its non-zero values are precisely its similarity factors; in particular, they form a subgroup of k^{×}. The most important examples of round quadratic forms are Pfister forms and quasi-Pfister forms [15, Cor. 9.9, Cor. 10.13].

7.2. Pointed quadratic spaces over arbitrary fields. Adopting the terminology of Weiss [55, Def. 1.1], by a pointed quadratic space over k we mean a triple Q = (V, q, e) consisting of a finite-dimensional vector space V over k, a quadratic form q: V → k and a vector e ∈ V which is a base point for q in the sense that q(e) = 1. Morphisms of pointed quadratic spaces are isometries preserving base points. Q is said to be non-singular (resp. anisotropic, round, Pfister, …) if q is. Given a conic algebra C over k, we obtain in Q_{C} := (C, n_{C}, 1_{C}) a pointed quadratic space and every pointed quadratic space arises in this manner (Loos [31], see also Rosemeier [44]). Notationally, we do not always distinguish carefully between C and Q_{C}. If Q = (V, q, e) is any pointed quadratic space over k, we therefore find it convenient to put V_{Q} = V as a vector space over k and to call n_{Q} := q the norm, 1_{Q} := e the unit element, t_{Q} := ∂n_{Q}(1_{Q}, −) the trace, ι_{Q}: Q → Q, x ↦ x^{∗} := t_{Q}(x)1_{Q} − x the conjugation of Q. We also put V_{Q}^{×} := {x ∈ Q | n_{Q}(x) ≠ 0}.

When dealing with quadratic forms representing 1, insisting on pointedness is not so much a matter of necessity but one of convenience, making the language of nonassociative algebras the natural mode of communication. Just as for composition algebras, Witt’s theorem [15, Thm. 8.3] implies that pointed quadratic spaces are classified by their norms:

7.3. Proposition. For non-singular pointed quadratic spaces over k to be isomorphic it is necessary and sufficient that their underlying quadratic forms be isometric. □

7.4. Enlargements. Let P be a pointed quadratic space over k and μ ∈ k^{×}. Writing V_{P} ⊕ V_{P} j for the direct sum of two copies of the vector space V_{P} over k as in 1.10, and identifying V_{P} ⊆ V_{P} ⊕ V_{P} j as a subspace through the first summand,

Q := ⟨⟨μ⟩⟩ ⊗ P := (V_{Q}, n_{Q}, 1_{Q}), V_{Q} := V_{P} ⊕ V_{P} j, n_{Q} := ⟨⟨μ⟩⟩ ⊗ n_{P} , 1_{Q} := 1_{P}

with

(1)  (⟨⟨μ⟩⟩ ⊗ n_{P} )(u + vj) = n_{P} (u) − μn_{P} (v)  (u, v ∈ V_{P})

is again a pointed quadratic space over k whose trace and conjugation are given by the formulas

(2)  t_{Q}(u + vj) = t_{P} (u),
(3)  (u + vj)^{∗} = u^{∗} − vj

for all u, v ∈ V_{P}. Moreover, if P is round (resp. Pfister), so is Q. Finally, a comparison with the Cayley-Dickson construction 1.10 shows Q_{Cay(B,μ)} = ⟨⟨μ⟩⟩⊗Q_{B} for any conic k-algebra B and any μ ∈ k^{×}. The following two statements are standard facts about round quadratic forms, translated into the setting of pointed quadratic spaces; we refer to [15, Prop. 9.8, Lemma 23.1] for details.

7.5. Proposition. Let P be a non-singular round pointed quadratic space over k.

(a) For μ ∈ k^{×}, the following conditions are equivalent.

(i) ⟨⟨μ⟩⟩ ⊗ P is isotropic.

(ii) μ ∈ n_{P} (V_{P}^{×}).

(iii) ⟨⟨μ⟩⟩ ⊗ P is hyperbolic.

(b) Let μ_{1}, μ_{2} ∈ k^{×}. Then ⟨⟨μ_{1}⟩⟩ ⊗ P ≅ ⟨⟨μ_{2}⟩⟩ ⊗ P if and only if μ_{1} = μ_{2}n_{P} (u) for some u ∈ P^{×}. □

7.6. Proposition. (Embedding property) Let Q be a pointed Pfister quadratic space over k and P ⊂ Q a proper pointed Pfister quadratic subspace. Then the inclusion P ↪ Q extends to an embedding from ⟨⟨μ⟩⟩ ⊗ P to Q, for some μ ∈ k^{×}. □

7.7. 2-Henselian fields. Let F be a field of arbitrary characteristic that is endowed with a normalized discrete valuation λ, so λ: F → ℤ_{∞} := ℤ ∪ {∞} is a surjective map satisfying the following conditions, for all α, β ∈ F.

λ is definite: λ(α) = ∞ ⟺ α = 0.
λ is sub-additive: λ(α + β) ≥ min {λ(α), λ(β)}.
λ is multiplicative: λ(αβ) = λ(α) + λ(β).

As convenient references for the theory of valuations we mention [14, 17], and particularly [32] for the discrete case. We write 𝔬 ⊆ F for the valuation ring of F relative to λ, 𝔭 ⊆ 𝔬 for its valuation ideal and F̄ := 𝔬/𝔭 for the residue field of F. The natural map from 𝔬 to F̄ will always be indicated by α ↦ ᾱ. Throughout the remainder of this paper, we fix a prime element π ∈ 𝔬. The quantity

e_{F} := λ(2 · 1_{F}), (1)

which is either a non-negative integer or ∞, will play an important role in the sequel. If F̄ has characteristic 2, then e_{F} > 0 agrees with what is usually called the absolute ramification index of F.

Due to the quadratic character of the gadgets we are interested in (composition and conic algebras, pointed quadratic spaces), requiring F to be Henselian (with respect to λ) is too strong a condition. It actually suffices to assume that F be 2-Henselian in the sense of Dress [12] or [17, §4.2], i.e., that F satisfies the following two equivalent conditions [12, Satz 1].

(i) For any quadratic field extension K/F, there is a unique extension of λ to a discrete valuation λ_{K} of K taking values in ℚ_{∞} = ℚ ∪ {∞}.

(ii) For all α_{0}, α_{1}, α_{2} ∈ 𝔬 with α_{0} ∈ 𝔭, α_{1} ∉ 𝔭, the polynomial

α_{0}t^{2} + α_{1}t + α_{2} ∈ F [t]

is reducible. In this case, the extension λ_{K} of λ in (i) is given by

(2)  λ_{K}(u) = \frac{1}{2}λ(N_{K/F} (u))  (u ∈ K).

From now on, F is assumed to be a fixed 2-Henselian field with respect to a normalized discrete valuation λ. For simplicity, all algebras, quadratic forms etc. over F are assumed to be finite-dimensional. The characteristic of F is arbitrary.

7.8. Quadratic forms over 2-Henselian fields. Let q: V → F be a quadratic form over F and suppose q is anisotropic. Following Springer [46] (in the case of a complete rather than Henselian valuation),

(1)  λ(q(x + y)) ≥ min {λ(q(x)), λ(q(y))}  (x, y ∈ V).

For convenience, we give the easy proof of this inequality. It evidently suffices to show

λ(q(x, y)) ≥ min {λ(q(x)), λ(q(y))} (2)

for all x, y ∈ V. Suppose there are x, y ∈ V such that (2) does not hold. Then q(x, y) ≠ 0, and the polynomial

\frac{1}{q(x, y)}q(tx + y) = \frac{q(x)}{q(x, y)}t^{2} + t + \frac{q(y)}{q(x, y)} ∈ F [t]

has no zero in F since q is anisotropic, but is reducible by 7.7 (ii), a contradiction. □

Relation (2) may be strengthened to

(3)  2λ(q(x, y)) ≥ λ(q(x)) + λ(q(y))  (x, y ∈ V),

either by appealing to a general result of Bruhat-Tits [8, Thm. 10.1.15], or by using an ad-hoc argument from the valuation theory of Jordan rings [40] in disguise: for x, y ∈ V, x ≠ 0, we obtain

Q_{x}y := q(x, y)x − q(x)y = −q(x)τ_{x}(y),

where τ_{x}: V → V is the reflection in the hyperplane perpendicular to x. In particular, τ_{x} leaves q invariant, which implies λ(q(Q_{x}y)) = 2λ(q(x)) + λ(q(y)), and since q(x, y)x = Q_{x}y + q(x)y, we obtain (3).

7.9. Valuation data for pointed quadratic spaces. For the rest of this section, we fix a pointed quadratic space Q over F which is round and anisotropic.

(a) The map

λ_{Q}: V_{Q} ⟶ ℚ_{∞}, x ⟼ λ_{Q}(x) := \frac{1}{2}λ(n_{Q}(x)), (1)

is a norm of V_{Q} as an F -vector space in the sense of Bruhat-Tits [9, 1.1], that is, the following relations hold for all α ∈ F, x, y ∈ V_{Q}.

(2)  λ_{Q} is definite: λ_{Q}(x) = ∞ ⟺ x = 0,
(3)  λ_{Q} is sub-additive: λ_{Q}(x + y) ≥ min {λ_{Q}(x), λ_{Q}(y)},
(4)  λ_{Q} is scalar-compatible: λ_{Q}(αx) = λ(α) + λ_{Q}(x),

where (3) is a consequence of (7.8.1). Moreover,

(5)  λ_{Q}(x^{∗}) = λ_{Q}(x),
(6)  λ(n_{Q}(x, y)) ≥ λ_{Q}(x) + λ_{Q}(y),
(7)  λ(t_{Q}(x)) ≥ λ_{Q}(x),

for all x, y ∈ V_{Q}, where (5) follows from conjugation invariance of n_{Q}, (6) from (7.8.3), and (7) from (6) for y = 1_{Q}.

(b) n_{Q} being round, Γ_{Q} := λ_{Q}(V_{Q}^{×}) is an additive subgroup of ℚ for which (1) implies

ℤ ⊆ Γ_{Q} ⊆ \frac{1}{2}ℤ, e_{Q/F} := [Γ_{Q} : ℤ] ∈ \{1, 2\}, Γ_{Q} = \frac{1}{e_{Q/F}} ℤ. (8)

We call e_{Q/F} the ramification index of Q.

(c) We put

(9)  𝔬_{Q} := {x ∈ V_{Q} | λ_{Q}(x) ≥ 0},
(10)  𝔭_{Q} := \{x ∈ V_{Q} | λ_{Q}(x) > 0\} = \{x ∈ V_{Q} | λ_{Q}(x) ≥ \frac{1}{e_{Q/F}} \} ⊆ 𝔬_{Q},

which are both full 𝔬-lattices in V_{Q}, and

𝔬_{Q}^{×} := 𝔬_{Q} \ 𝔭_{Q} = {x ∈ V_{Q} | λ_{Q}(x) = 0}, (11)

which is just a subset of 𝔬_{Q} containing 1_{Q}. By abuse of language, the elements of 𝔬_{Q}^{×} are called units of 𝔬_{Q}. An element Π ∈ V_{Q}^{×} such that λ_{Q}(Π) > 0 generates the infinite cyclic group Γ_{Q} belongs to 𝔭_{Q} and is called a prime element of 𝔬_{Q}. Writing x ↦ x̄ for the natural map from 𝔬_{Q} to 𝔬_{Q}/𝔭_{Q} and setting

Q̄ := (VQ̄, nQ̄, 1Q̄), VQ̄ := 𝔬_{Q}/𝔭_{Q}, 1Q̄ := \overline{1_{Q}},

where nQ̄: Q̄ ⟶ F̄, x̄ ⟼ nQ̄(x̄) := \overline{n_{Q}(x)} is the first residue form of n_{Q}, we obtain a pointed quadratic space over F̄, called the pointed quadratic residue space of Q, which is round and anisotropic. Here only roundness of nQ̄ demands a proof, so let u ∈ 𝔬_{Q}^{×}. Since n_{Q} is round, there exists a linear bijection f: V_{Q} → V_{Q} with n_{Q}(f(x)) = n_{Q}(u)n_{Q}(x) for all x ∈ V_{Q}. Hence f stabilizes 𝔬_{Q} as well as 𝔭_{Q} and thus canonically induces a similarity transformation VQ̄ → VQ̄ relative to nQ̄ with multiplier nQ̄(ū).

We call

f_{Q/F} := dimF̄(VQ̄) (12)

the residue degree of Q. For convenience, we collect some of the above properties of Q̄ in the following proposition, which is really stating the obvious.

7.10. Proposition. Norm, trace and conjugation of the pointed quadratic residue space Q̄ of Q are given by the formulas

(1)  nQ̄(x̄) = \overline{n_{Q}(x)},
(2)  tQ̄(x̄) = \overline{t_{Q}(x)},
(3)  x̄^{∗} = \overline{x^{∗}}

for all x ∈ 𝔬_{Q}. Moreover, nQ̄ is round and anisotropic. □

Here is the easiest example that is not totally trivial.

7.11. Example. Let Q = (V, q, e) be such that V has basis e, j and q(ue + vj) = u^{2} − μv^{2} for some μ ∈ 𝔬^{×} and all u, v ∈ F. (This is the case P = ⟨1⟩ of 7.4; q ≅ ⟨⟨μ⟩⟩.) If μ̄ is not a square in \overline{F}, then 𝔬_{Q} and 𝔭_{Q} are the 𝔬- and 𝔭-spans of e, j respectively and the pointed quadratic residue space of Q is naturally identified with the quadratic extension \overline{F} (\sqrt{μ̄}) with quadratic form the squaring map z ↦ z^{2} and base point 1 ∈ \overline{F}.

7.12. Lemma. The map

ℤ × \{y ∈ V_{Q}^{×} | 0 ≤ λ_{Q}(y) ≤ 1 − \frac{1}{e_{Q/F}} \} ⟶ V_{Q}^{×}, (m, y) ⟼ π^{m}y,

is surjective.

Proof. For x ∈ V_{Q}^{×}, some n ∈ ℤ has λ_{Q}(x) = \frac{n}{e_{Q/F}} by (7.9.8), and writing n =

me_{Q/F} + r, m, r ∈ ℤ, 0 ≤ r ≤ e_{Q/F} − 1, the element y := π^{−m}x ∈ V_{Q}^{×} satisfies 0 ≤ λ_{Q}(y) ≤ 1 − \frac{1}{e_{Q/F}} .

□

7.13. Proposition. e_{Q/F} f_{Q/F} = dim_{F} (V_{Q}).

Proof. If e_{Q/F} = 1, then 𝔭_{Q} = π𝔬_{Q}, and the dimension of VQ̄ = 𝔬_{Q} ⊗_{𝔬} F̄ over F̄ agrees with the dimension of V_{Q} over F. We may therefore assume e_{Q/F} = 2. Let Π be a prime element of 𝔬_{Q} and f: V_{Q} → V_{Q} a norm similarity with multiplier n_{Q}(Π). Then λ_{Q}(f(x)) = \frac{1}{2} + λ_{Q}(x) for all x ∈ V_{Q}^{×}, which implies f(𝔬_{Q}) = 𝔭_{Q}, f(𝔭_{Q}) = π𝔬_{Q}, and f induces canonically an F̄-linear bijection from VQ̄ onto 𝔭_{Q}/π𝔬_{Q}. Combined with the filtration 𝔬_{Q} ⊗_{𝔬} F̄ = 𝔬_{Q}/π𝔬_{Q} ⊃ 𝔭_{Q}/π𝔬_{Q} ⊃ {0} and the isomorphism (𝔬_{Q}/π𝔬_{Q})/(𝔭_{Q}/π𝔬_{Q}) ≅ VQ̄ as vector spaces over F̄, this implies dim_{F} (V_{Q}) = 2dimF̄(V_{Q̄)}, as desired. □

Remark. Prop. 7.13 becomes false if Q is not assumed to be round, for example if e_{Q/F} = 2 and dim_{F} (V_{Q}) is odd, see [40, Satz 6.3] for generalization.

7.14. Connecting with conic algebras. Let C be a finite-dimensional conic algebra over F and suppose its norm is round and anisotropic; for example, C could itself be a composition algebra, or it could arise from a composition algebra by means of the Cayley-Dickson process. Applying as we may the preceding considerations to Q_{C}, the pointed quadratic space associated with C via 7.2, we systematically adhere to the following convention: all notation and terminology developed up to now and later on for pointed quadratic spaces will be applied without further comment to C in place of Q_{C}, modifying subscripts accordingly whenever possible. For example,

λ_{C}: C ⟶ ℚ_{∞}, x ⟼ λ_{C}(x) := \frac{1}{2}λ(n_{C}(x)), (1)

is a norm of C as an F -vector space, e_{C/F} := e_{Q_{C}/F} is the ramification index, f_{C/F} := f_{Q_{C}/F} is the residue degree and C̄ := \overline{Q_{C}} is the pointed quadratic residue space of C. But note that C̄ in general is not a conic algebra over F̄ in a natural way unless C is a composition algebra; see 7.15 and Section 10 below for further discussion.

7.15. Tame and wild pointed quadratic spaces. If tQ̄ is non-zero, then Q is said to be tame. Otherwise, i.e., if tQ̄ = 0, then Q is said to be wild. For Q to be wild it is clearly necessary that F̄ have characteristic 2. Applying [41, Prop. 1], and bearing in mind the conventions of 7.14, a composition division algebra C over F is tame (resp. wild) iff C̄ is a composition algebra (resp. a purely inseparable field extension of exponent at most 1) over F̄. Extending the terminology of [41] to the present more general set-up, we call Q unramified (resp. ramified) if Q is tame with e_{Q/F} = 1 (resp. e_{Q/F} = 2). For Q to be unramified it is necessary and sufficient that the quadratic form n_{Q} have good reduction with respect to λ in the sense of Knebusch [27, 28]. This will follow from Prop. 8.2 (c) below.

The preceding definitions seem to assign a distinguished role to the base point of a pointed quadratic space. But this is not so as will be seen in Prop. 8.2 below.

8. Trace and norm exponent.

This section serves a double purpose. Working with a fixed non-singular, round and anisotropic pointed quadratic space Q over a 2-Henselian field F, we attach wildness-detecting invariants to Q. Moreover, we present a device measuring how far a scalar in 𝔬^{×} is removed from being the norm of an appropriate element in V_{Q}.

8.1. Trace ideal and trace exponent. Since 𝔬_{Q} ⊆ V_{Q} is a full 𝔬-lattice, its image under the trace of Q by non-singularity is a non-zero ideal in 𝔬, called the trace ideal of Q. But 𝔬 is a discrete valuation ring, so there is a unique integer texp(Q) ≥ 0 such that

t_{Q}(𝔬_{Q}) = 𝔭^{texp(Q)}. (1)

We call texp(Q) the trace exponent of Q and have

texp(Q) = min {λ(t_{Q}(x)) | x ∈ 𝔬_{Q}}. (2)

The image of 𝔬_{Q} ⊗_{𝔬} 𝔬_{Q} under the linear map x ⊗ y ↦ n_{Q}(x, y) is an ideal in 𝔬 denoted by ∂n_{Q}(𝔬_{Q} ⊗_{𝔬} 𝔬_{Q}). The following result relates the ideals just defined to one another but also to wild and tame pointed quadratic spaces.

8.2. Proposition. (a) Q is wild if and only if texp(Q) > 0.

(b) If P ⊆ Q is pointed quadratic subspace that is round and non-singular, then texp(P) ≥ texp(Q).

(c) t_{Q}(𝔬_{Q}) = ∂n_{Q}(𝔬_{Q} ⊗_{𝔬} 𝔬_{Q}).

(d) Q is tame if and only if Q̄ is non-singular.

Proof. (a) and (b) are obvious. Before proving (c),(d), we let x ∈ Q^{×} and, by using a Jordan isotopy argument in disguise, pass to Q^{x} := (V_{Q}, n_{Q}(x)^{−1}n_{Q}, x), which is a non-singular quadratic space over F. Since n_{Q} is round, the norms of Q and Q^{x} are isometric, forcing Q and Q^{x} to be isomorphic as pointed quadratic spaces by Prop. 7.3. This implies

(1)  t_{Q}(𝔬_{Q}) = t_{Q^{x}}(𝔬_{Q^{x}}) = {n_{Q}(x, n_{Q}(x)^{−1}y) | y ∈ V_{Q}, λ_{Q}(y) ≥ λ_{Q}(x)}
= {n_{Q}(x, y) | y ∈ V_{Q}, λ_{Q}(y) ≥ −λ_{Q}(x)}.

(c) The left-hand side is clearly contained in the right, so it suffices to show n_{Q}(x, y) ∈ t_{Q}(𝔬_{Q}) for all x, y ∈ 𝔬_{Q}, x ≠ 0. But this follows from (1) and λ_{Q}(x) ≥ 0.

(d) Non-singularity of Q̄ is clearly sufficient for Q to be tame. Conversely, suppose Q is tame and let x ∈ 𝔬_{Q}^{×}. Then (1) produces an element y ∈ 𝔬_{Q} with n_{Q}(x, y) = 1 ∈ t_{Q}(𝔬_{Q}), hence nQ̄(x̄, ȳ) ≠ 0. □

8.3. Trace generators. An element w_{0} ∈ 𝔬_{Q} where the minimum in (8.1.2) is attained, i.e., with λ(t_{Q}(w_{0})) = texp(Q) is called a trace generator of Q. If even t_{Q}(w_{0}) = π^{texp(Q)}, we speak of a normalized trace generator, dependence on π being understood. Trace generators always exist (as do normalized ones) and their traces generate the trace ideal of C. Moreover, they satisfy the inequalities

0 ≤ λ_{Q}(w_{0}) ≤ 1 − \frac{1}{e_{Q/F}} . (1)

Indeed, assuming λ_{Q}(w_{0}) > 1 − \frac{1}{e_{Q/F}} implies

λ_{Q}(π^{−1}w_{0}) = λ_{Q}(w_{0}) − 1 > −\frac{1}{e_{Q/F}} ,

and since λ_{Q}(π^{−1}w_{0}) belongs to \frac{1}{e_{Q/F}} ℤ, we conclude π^{−1}w_{0} ∈ 𝔬_{Q}. But now

t_{Q}(π^{−1}w_{0}) = π^{texp(Q)−1} ∉ 𝔭^{texp(Q)} leads to a contradiction.

8.4. Tignol’s invariant ω(C). Another invariant that fits into the present set-up is due to Tignol [51, pp. 9,17] for separable field extensions and central associative division algebras of degree p over Henselian fields (not necessarily discrete) having residual characteristic p > 0. As a straightforward adaptation of Tignol’s definition to the situation we are interested in, we define

ω(Q) := min {λ(t_{Q}(x)) − λ_{Q}(x) | x ∈ V_{Q}^{×}}; (1)

thanks to (7.9.7), it is a non-negative rational number. Moreover, it is closely related to the trace exponent, as the following proposition shows.

8.5. Proposition. (a) ω(Q) = texp(Q) or ω(Q) = texp(Q) − \frac{1}{2}.

(b) There exists a trace generator w_{0} of Q with

ω(Q) = texp(Q) − λ_{Q}(w_{0}). (1)

(c) If e_{Q/F} = 1, then ω(Q) = texp(Q).

Proof. Since the map

φ: V_{Q}^{×} ⟶ Γ_{Q} = \frac{1}{e_{Q/F}} ℤ, x ⟼ φ(x) := λ(t_{Q}(x)) − λ_{Q}(x),

is homogeneous of degree zero, so φ(αx) = φ(x) for all α ∈ F^{×}, x ∈ V_{Q}^{×}, Lemma 7.12 implies ω(Q) = min {φ(x) | x ∈ S}, where

S := \{x ∈ V_{Q}^{×} | 0 ≤ λ_{Q}(x) ≤ 1 − \frac{1}{e_{Q/F}} \}.

Accordingly, let w_{0} ∈ S satisfy

φ(w_{0}) = ω(Q). (2)

Given any trace generator w′_{0} of Q, the chain of inequalities

(3) 0 ≤ λ_{Q}(w′_{0}) = λ(t_{Q}(w′_{0})) − φ(w′_{0}) ≤ texp(Q) − ω(Q) ≤ λ(t_{Q}(w_{0})) − φ(w_{0})
= λ_{Q}(w_{0}) ≤ 1 − \frac{1}{e_{Q/F}}

implies (a) and (c), while in (b) we may assume e_{Q/F} = 2 since otherwise (1) holds for w′_{0}, i.e., for any trace generator of Q. Since all quantities in (3) belong to \frac{1}{2}ℤ, we either have λ_{Q}(w′_{0}) = texp(Q) − ω(Q) or

texp(Q) − ω(Q) = λ(t_{Q}(w_{0})) − φ(w_{0}) = λ_{Q}(w_{0}) = \frac{1}{2}.

In the latter case, (2) shows that w_{0} is a trace generator of Q, and the proof of (b) is complete. □

8.6. Regular trace generators. Trace generators of Q satisfying (8.5.1) are called regular. If Q has ramification index 1, then every trace generator by (8.3.1) and Prop. 8.5 (c) is regular but, in general, this need not be so, cf. Cor. 11.7 and Thm. 12.3 below.

8.7. Example. If F has characteristic not 2, it is a composition division algebra over itself, with norm and trace given by n_{F} (α) = α^{2}, t_{F} (α) = 2α for all α ∈ F. Hence (7.7.1), (8.1.1) and (8.4.1) imply

texp(F) = ω(F) = e_{F} , (1)

and w_{0} = \frac{π^{e_{F}}}{2} ∈ 𝔬^{×} is the unique normalized trace generator of F; it is obviously regular. From (1) and Props. 8.2 (c), 8.5 we now conclude

0 ≤ ω(Q) ≤ texp(Q) ≤ e_{F} , (2)

which for trivial reasons also holds in characteristic 2.

8.8. The norm exponent. We wish to measure how far a given unit in the valuation ring of F is removed from being the norm of an element in V_{Q} or, equivalently, in 𝔬_{Q}^{×}. To this end, we observe that, given α ∈ 𝔬^{×} and an integer d ≥ 0, the following conditions are equivalent.

(i) α is a norm of 𝔬_{Q}^{×} mod 𝔭^{d}, i.e., there exists an element v ∈ 𝔬_{Q}^{×} with
α − n_{Q}(v) ∈ 𝔭^{d}.
(ii) There exist elements β ∈ 𝔬, v ∈ 𝔬_{Q}^{×} with
α = (1 − π^{d}β)n_{Q}(v).

Thus the set N_{Q}(α) of non-negative integers d satisfying (i)/(ii) above contains 0, and d ∈ N_{Q}(α) implies d′ ∈ N_{Q}(α) for all integers d′, 0 ≤ d′ ≤ d. We therefore put

nexp_{Q}(α) := sup N_{Q}(α) (1)

and call this the norm exponent of α relative to Q; it is either a non-negative integer or ∞. Roughly speaking, the bigger the norm exponent becomes, the closer α gets to being a norm of Q. More precisely, 2^{− nexp_{Q}(α)} is the (minimum) distance of α from the subset n_{Q}(𝔬_{Q}^{×}) ⊆ F^{×} relative to the metric induced by the absolute value ξ ↦ |ξ| := 2^{−λ(ξ)}.

A number of useful elementary properties of the norm exponent are collected in the following proposition, whose straightforward proof is left to the reader.

8.9. Proposition. With the notations and assumptions of 8.8, let α, α′ ∈ 𝔬^{×}.

(a) If α ∈ n_{Q}(𝔬_{Q}^{×}), then nexp_{Q}(α) = ∞.

(b) If d = nexp_{Q}(α) is finite, then α can be written in the form

α = (1 − π^{d}β)n_{Q}(v), β ∈ 𝔬, v ∈ 𝔬_{Q}^{×},

and every such representation of α satisfies β ∈ 𝔬^{×}.

(c) nexp_{Q}(α) = 0 if and only if ᾱ ∉ nQ̄(V × ; Q̄).

(d) α ≡ α′ mod n_{Q}(V_{Q}^{×}) implies nexp_{Q}(α) = nexp_{Q}(α′).

(e) nexp_{P} (α) ≤ nexp_{Q}(α) for any round non-singular pointed quadratic subspace P ⊆ Q.

□

By Prop. 8.9 (a), the elements of n_{Q}(𝔬_{Q}^{×}) have infinite norm exponent. While the converse is also true, we can do better than that by showing that the norm exponents of elements in 𝔬^{×} \ n_{Q}(𝔬_{Q}^{×}) are uniformly bounded from above. Indeed, we have the following result.

8.10. Local Norm Theorem. Let α ∈ 𝔬^{×} \ n_{Q}(𝔬_{Q}^{×}) = 𝔬^{×} \ n_{Q}(V_{Q}^{×}). Then

nexp_{Q}(α) ≤ 2 ω(Q).

More precisely, letting w_{0} ∈ 𝔬_{Q} be a normalized regular trace generator of Q, every β ∈ 𝔬 admits a γ ∈ 𝔬 with

1 − π^{2 ω(Q)+1}β = n_{Q}(1_{Q} + π^{texp(Q)}γw_{0}). (1)

Proof. Arguing indirectly, and using roundness of n_{Q}, the first part of the theorem follows from the second. To establish the second part, we apply Prop. 8.5 (a),(b) and obtain

0 ≤ d := λ(n_{Q}(w_{0})) = 2λ_{Q}(w_{0}) = 2( texp(Q) − ω(Q)) ≤ 1.

Since F is 2-Henselian, the polynomial

g := π^{1−d}n_{Q}(w_{0})t^{2} + t + β ∈ 𝔬[t] ⊆ F [t]

by 7.7 (ii) is reducible, and the two roots δ, δ′ ∈ F of π^{d−1}n_{Q}(w_{0})^{−1}g satisfy the relation δδ′ = π^{d−1}n_{Q}(w_{0})^{−1}β ∈ 𝔭^{−1}. Thus we may assume δ ∈ 𝔬 and have π^{1−d}n_{Q}(w_{0})δ^{2} + δ + β = 0. Setting γ := π^{1−d}δ ∈ 𝔬, we therefore obtain

n_{Q}(1 + π^{texp(Q)}γw_{0}) = 1 + π^{texp(Q)+1−d}δt_{Q}(w_{0}) + π^{2 texp(Q)}π^{2(1−d)}δ^{2}n_{Q}(w_{0})
= 1 + π^{2 texp(Q)−d+1}(δ + π^{1−d}n_{Q}(w_{0})δ^{2})
= 1 − π^{2 ω(Q)+1}β.  □

For a version of this result addressed to central associative division algebras of degree p = char(F̄) > 0, see Kato [23, Prop. 2 (iii)].

8.11. Corollary. Let μ, μ′ ∈ 𝔬^{×}. Then

μ ≡ μ′ mod n_{Q}(Q^{×}) ⟺ ∃ x ∈ 𝔬_{Q}^{×} : μ ≡ μ′n_{Q}(x) mod 𝔭^{2 ω(Q)+1}.

□

8.12. The connection with the quadratic defect. We assume char(F) ≠ 2 and return to the composition division algebra C = F of 8.7. Comparing 8.8 with [35, §63A], we see that 𝔭^{nexp_{F} (α)} is the quadratic defect of α ∈ 𝔬^{×}. Moreover, texp(F) = ω(F) = e_{F}, and w_{0} := \frac{π^{e_{F}}}{2} ∈ 𝔬^{×} is the unique normalized regular trace generator of F. In particular, \frac{4}{π^{2e_{F}}} ∈ 𝔬^{×}, and the change of variables β = −\frac{4}{π^{2e_{F}}} β′, γ = \frac{4}{π^{2e_{F}}} γ′ converts (8.10.1) into the relation

1 + 4πβ′ = (1 + 2πγ′)^{2}.

Hence the local norm theorem becomes the local square theorem of [35, 63:1] or [30, VI.2.19] in the special case Q = Q_{F}. See also [10, Prop. 4.1.2] for an extension of this result to residual characteristics other than 2.

The local norm theorem has numerous applications. One of these can already be given in this section; a useful technical lemma prepares the way.

8.13. Lemma. Suppose Q is wild and has ramification index e_{Q/F} = 1. For d ∈ ℤ, 0 ≤ d ≤ 2 texp(Q) = 2 ω(Q) and β ∈ 𝔬^{×}, the following conditions are equivalent.

(i) 1 − π^{d}β ∈ n_{Q}(𝔬_{Q}^{×}).
(ii) 1 − π^{d}β ∈ 𝔬^{×} and there are elements m ∈ ℤ, w ∈ 𝔬_{Q}^{×} with

d = 2m, β = −n_{Q}(w) + π^{−m}t_{Q}(w). (1)

Proof. (i) ⟹ (ii). There exists an element v ∈ 𝔬_{Q}^{×} with 1−π^{d}β = n_{Q}(v) ∈ 𝔬^{×}. For d = 0 we put w = 1_{Q} −v ∈ 𝔬_{Q} and obtain 1−β = n_{Q}(1_{Q} −w) = 1−t_{Q}(w)+n_{Q}(w), hence β = −n_{Q}(w) + t_{Q}(w). But t_{Q}(w) ∈ 𝔭^{texp(Q)} ⊆ 𝔭 since Q is wild, forcing w ∈ 𝔬_{Q}^{×}. Thus (ii) holds with m = 0. We may therefore assume d > 0. Then 1 = \overline{n_{Q}(v)} = nQ̄(v̄), forcing v̄ = 1 since nQ̄ is anisotropic and nQ̄(1Q̄ − v̄) = 0 by wildness of Q. Combining with e_{Q/F} = 1, we find an integer m > 0 and a unit w ∈ 𝔬_{Q}^{×} with v = 1_{Q} −π^{m}w. Expanding the right-hand side of 1−π^{d}β = n_{Q}(1_{Q} −π^{m}w), we conclude

π^{d}β = −π^{2m}n_{Q}(w) + π^{m}t_{Q}(w), (2)

which in turn yields the estimate

2 texp(Q) ≥ d = λ(π^{d}β) = λ( − π^{2m}n_{Q}(w) + π^{m}t_{Q}(w)) ≥ min {2m, m + texp(Q)}.

This implies

m ≤ texp(Q), d ≥ 2m, (3)

and (2) attains the form

π^{d}β = −π^{2m}n_{Q}(w)(1 − γ), γ := π^{−m}n_{Q}(w)^{−1}t_{Q}(w) ∈ 𝔭^{texp(Q)−m}. (4)

If m < texp(Q), then (1) follows from (2),(4). On the other hand, if m = texp(Q), then (3) implies d = 2 texp(Q) = 2m, and (1) follows from (2).

(ii) ⟹ (i). Setting v := 1_{Q} − π^{m}w ∈ 𝔬_{Q} and applying (1), we conclude n_{Q}(v) = 1 − π^{d}β ∈ 𝔬^{×}, hence v ∈ 𝔬_{Q}^{×}, and (i) holds. □

Remark. For d > 0, the condition 1 − π^{d}β ∈ 𝔬^{×} in (ii) is of course automatic.

8.14. Proposition. Let P be a pointed quadratic space over F that is non-singular, round and anisotropic with e_{P/F} = 1, and let d be an odd integer with 0 ≤ d < 2 texp(P).

(a) If β ∈ 𝔬, then μ := 1 − π^{d}β ∈ 𝔬^{×} and

d = nexp_{P} (μ) ⟺ β ∈ 𝔬^{×}.

(b) If β_{i} ∈ 𝔬^{×} and μ_{i} := 1 − π^{d}β_{i} for i = 1, 2, then

⟨⟨μ_{1}⟩⟩ ⊗ P ≅ ⟨⟨μ_{2}⟩⟩ ⊗ P ⟹ \overline{β_{1}} = \overline{β_{2}}.

Proof. (a) d = nexp_{P} (μ) implies β ∈ 𝔬^{×} by Prop. 8.9 (b). Conversely, suppose β ∈ 𝔬^{×}. Then μ ∉ n_{P} (V_{P}^{×}) by Lemma 8.13. Thus the local norm theorem 8.10 implies d ≤ d′ := nexp_{P} (μ) ≤ 2 texp(P), and from Prop. 8.9 (b) we obtain a representation μ = μ′n_{P} (v′), μ′ = 1 − π^{d′}β′ for some β′ ∈ 𝔬^{×}, v′ ∈ 𝔬_{P}^{×}. Assuming d < d′ would imply that μμ′ = 1 − π^{d}γ, γ := β + π^{d′−d}β′ − π^{d′}ββ′ ∈ 𝔬^{×}, does not belong to n_{P} (V_{P}^{×}), again by Lemma 8.13, in contradiction to μμ′ = n_{P} (μ′v′).

(b) Arguing indirectly, let us assume \overline{β_{1}} ≠ \overline{β_{2}}. Then μ_{1}μ_{2} = 1 − π^{d}β, where β = β_{1} + β_{2} − π^{d}β_{1}β_{2} ∈ 𝔬 satisfies β̄ = \overline{β_{1}} − \overline{β_{2}} ≠ 0, hence β ∈ 𝔬^{×}. By (a), the norm exponent of μ_{1}μ_{2} relative to P is d, forcing μ_{1}μ_{2} ∉ n_{P} (V_{P}^{×}) (this also follows from Lemma 8.13). Hence ⟨⟨μ_{1}⟩⟩ ⊗ P and ⟨⟨μ_{2}⟩⟩ ⊗ P are not isomorphic by Prop. 7.5 (b). □

We will see in Example 9.11 (b) below that the converse of Prop. 8.14 (b) does not hold.

8.15. Lemma. Let β ∈ 𝔬, w ∈ 𝔬_{Q} and suppose d, m are non-negative integers. Determine α ∈ 𝔬 by t_{Q}(w) = π^{texp(Q)}α. Then

(1 − π^{d}β)n_{Q}(1_{Q} − π^{m}w) = 1 − π^{d}β + π^{2m}n_{Q}(w) − π^{texp(Q)+m}α +
π^{texp(Q)+d+m}αβ − π^{d+2m}βn_{Q}(w).

Proof. Expand the left-hand side in the obvious way. □

9. Valuation data under enlargements.

In this section we will be concerned with the question of what happens to the valuation data ramification index (7.9 (b)), pointed quadratic residue space (7.9 (c)) and trace exponent (8.1) when passing from a pointed quadratic space P to ⟨⟨μ⟩⟩⊗P, μ ∈ F^{×}. We will answer this question not in full generality but only under the additional hypothesis that P have ramification index 1. This hypothesis derives its justification from the fact that, if F has characteristic zero, every anisotropic pointed (n+1)-Pfister quadratic space over F contains a pointed n-Pfister quadratic subspace of ramification index 1. We will prove this in Prop. 17.2 and Thm. 19.2(i) below by using methods from algebraic K-theory. It would be interesting to know whether the result in question also holds for F having characteristic 2.

9.1. The general set-up. (a) We fix a 2-Henselian field F and a pointed quadratic space P over F which is non-singular, round and anisotropic. We also assume throughout that P has ramification index e_{P/F} = 1, which implies

ω(P) = texp(P) (1) by Prop. 8.5 (c) and

Γ_{P} = λ_{P} (P^{×}) = ℤ, λ(n_{P} (P^{×})) = 2ℤ (2)

by 7.9 (b) and (7.9.1).

(b) We are interested in pointed quadratic spaces Q = ⟨⟨μ⟩⟩ ⊗ P, μ ∈ F^{×}, as in 7.4; in particular, we recall V_{Q} = V_{P} +V_{P} j as vector spaces over F. By Prop. 7.5 (b), we may and always will assume λ(μ) ∈ {0, 1}, so μ is either a unit or a prime element in 𝔬.

There are two harmless cases which we treat first. One of them arises when μ is a prime, the other when μ is a unit and P is tame.

9.2. Proposition. If μ is a prime element in 𝔬, then Q := ⟨⟨μ⟩⟩ ⊗ P is a nonsingular, round and anisotropic pointed quadratic space over F with

(1)  λ_{Q}(u + vj) = min \{λ_{P} (u), λ_{P} (v) + \frac{1}{2}\}  (u, v ∈ V_{P}),
(2)  𝔬_{Q} = 𝔬_{P} ⊕ 𝔬_{P} j, 𝔭_{Q} = 𝔭_{P} ⊕ 𝔬_{P} j,
(3)  e_{Q/F} = 2, Q̄ = P̄, texp(Q) = texp(P).

Proof. From (9.1.2) we conclude μ ∉ n_{P} (P^{×}). Thus Q is not only round and non-singular but also anisotropic, so λ_{Q} satisfying (7.9.1)−(7.9.4) exists. Since λ_{Q}(vj) = λ_{P} (v) + \frac{1}{2} ≠ λ_{P} (u) for all u, v ∈ V_{P}^{×}, by (7.4.1),(7.9.1) and again by (9.1.2), we obtain (1), hence (2) and the first two relations of (3), while the last one is immediately implied by (7.4.2). □

9.3. Proposition. If P as in 9.1 (a) is tame and μ ∈ 𝔬^{×}, then Q = ⟨⟨μ⟩⟩ ⊗ P is a non-singular and round pointed quadratic space over F. Moreover the following conditions are equivalent.

(i) Q is anisotropic. (ii) μ ∉ n_{P} (V_{P}^{×}).

(iii) μ̄ ∉ nP̄(V × ; P̄).

In this case,

(1)  λ_{Q}(u + vj) = min {λ_{P} (u), λ_{P} (v)}  (u, v ∈ P),
(2)  𝔬_{Q} = 𝔬_{P} ⊕ 𝔬_{P} j, 𝔭_{Q} = 𝔭_{P} ⊕ 𝔭_{P} j,

and Q is tame of ramification index e_{Q/F} = 1 with Q̄ ≅ ⟨⟨μ̄⟩⟩ ⊗ P̄.

Proof. While the first statement is obvious, the equivalence of (i),(ii),(iii) follows from Prop. 7.5 (a) and Cor. 8.11 since ω(P) = texp(P) = 0 by (9.1.1) and tameness of P. If (i),(ii),(iii) hold, then λ_{Q} exists and it suffices to show that (1) holds. Since λ_{Q}(vj) = λ_{P} (v) by (7.4.1),(7.9.1), we certainly have λ_{Q}(u + vj) ≥ min {λ_{P} (u), λ_{P} (v)}. To prove equality, Lemma 7.12 allows us to assume λ_{P} (u) = λ_{P} (v) = 0. Here λ_{Q}(u + vj) > 0 would imply nP̄(ū) = μ̄nP̄(v̄), hence μ̄ ∈ nP̄(P̄^{×}) since nP̄ is round, and we obtain a contradiction to (iii). □

The remaining cases where P is wild and μ ∈ 𝔬 is a unit are much more troublesome.

9.4. Some easy reductions. For the rest of this section, we assume that P as given in 9.1 (a) is wild, so ω(P) = texp(P) > 0 by Prop. 8.2 (a). We are interested in the pointed quadratic spaces ⟨⟨μ⟩⟩ ⊗ P, μ ∈ 𝔬^{×}, only when they are anisotropic. By Prop. 7.5 (a) and the local norm theorem 8.10, this is equivalent to μ having norm exponent ≤ 2 texp(P), so by Prop. 8.9 (b) it will be enough to consider units in 𝔬 having the form μ = (1 − π^{d}β)n_{P} (v), where d ∈ ℤ satisfies 0 ≤ d ≤ 2 texp(P), β ∈ 𝔬^{×} and v ∈ 𝔬_{P}^{×}. Here Prop. 7.5 (b) allows us to assume v = 1_{P}. We are thus reduced to working with scalars μ that may be written as

μ = 1 − π^{d}β, d ∈ ℤ, 0 ≤ d ≤ 2 texp(P), β ∈ 𝔬^{×}. (1)

Setting

0 ≤ m := ⌊\frac{d}{2}⌋ ≤ texp(P), Θ_{d} := π^{−m}(1_{P} + j) ∈ V_{Q} = V_{P} + V_{P} j, (2)

we define, inspired by the Cayley-Dickson construction of conic algebras (cf. (1.10.1)),

vΘ_{d} := π^{−m}(v + vj) (v ∈ V_{P}), V_{P} Θ_{d} := {vΘ_{d} | v ∈ V_{P} } (3)

and obtain after a straightforward computation, involving (7.4.1),(7.4.2),

(4)  V_{Q} = V_{P} ⊕ V_{P} Θ_{d},
(5)  n_{Q}(u + vΘ_{d}) = n_{P} (u) + π^{−m}n_{P} (u, v) + π^{d−2m}βn_{P} (v),
(6)  t_{Q}(u + vΘ_{d}) = t_{P} (u) + π^{−m}t_{P} (v)

for all v ∈ V_{P}.

9.5. Pointed quadratic residue spaces and inseparable extensions. (cf. [15, Remark 10.4]) For P as in 9.1,9.4, we claim that VP̄ carries a unique structure of a purely inseparable extension field over F̄ having exponent at most 1 such that nP̄(u′) = u^{′2} for all u′ ∈ VP̄. To see this, it suffices to note that nP̄ is round and anisotropic with ∂nP̄ = 0, making nP̄(VP̄) a subfield of F̄, so an F̄-bilinear multiplication VP̄ × VP̄ → VP̄, (u′, v′) ↦ u′v′, gives a purely inseparable extension field structure as indicated iff nP̄(u′v′) = nP̄(u′)nP̄(v′) for all u′, v′ ∈ VP̄. The purely inseparable extension field thus constructed will again be denoted by VP̄ if there is no danger of confusion.

9.6. Proposition. Let d be an odd integer with 0 ≤ d ≤ 2 texp(P) and β ∈ 𝔬^{×}. Then

μ := 1 − π^{d}β ∈ 𝔬^{×}

and Q := ⟨⟨μ⟩⟩ ⊗ P is a non-singular, round and anisotropic pointed quadratic space over F. Moreover, Q is wild and

Π := Θ_{d} = π^{− \frac{d−1}{2}} (1_{B} + j) ∈ V_{Q} (1) is a prime element of 𝔬_{Q} with

(2)  n_{Q}(Π) = πβ, t_{Q}(Π) = 2π^{− \frac{d−1}{2}} , V_{Q} = V_{P} ⊕ V_{P} Π,
(3)  λ_{Q}(u + vΠ) = min \{λ_{P} (u), λ_{P} (v) + \frac{1}{2}\} (u, v ∈ V_{P}),
(4)  𝔬_{Q} = 𝔬_{P} ⊕ 𝔬_{P} Π, 𝔭_{Q} = 𝔭_{P} ⊕ 𝔬_{P} Π,
(5)  e_{Q/F} = 2, Q̄ = P̄, texp(Q) = texp(P) − \frac{d − 1}{2}.

Proof. By Prop. 8.14 (a), nexp_{P} (μ) = d is finite, forcing Q to be anisotropic. Applying (9.4.4)−(9.4.6), we obtain (2); in particular, λ_{Q}(Π) = \frac{1}{2}, so Π is a prime element of 𝔬_{Q} and the first formula of (5) holds. We proceed to establish (3). If u, v ∈ V_{P}^{×}, then λ_{Q}(u) = λ_{P} (u) is an integer by (9.1.2), while λ_{Q}(vΠ) = λ_{P} (v) + \frac{1}{2} is not. This not only proves (3) but also (4) and the second formula of (5). The last one follows from the fact that (9.4.6) establishes 𝔭^{texp(P)} + 𝔭^{texp(P)− \frac{d−1}{2}} = 𝔭^{texp(P)− \frac{d−1}{2}} as the trace ideal of Q. □

9.7. Proposition. Let d be an even integer with 0 ≤ d < 2 texp(P) and suppose β ∈ 𝔬 satisfies the condition β̄ ∉ V 2 ; P̄ (cf. 9.5). Then

μ = 1 − π^{d}β ∈ 𝔬^{×} (1)

and Q := ⟨⟨μ⟩⟩ ⊗ P is a non-singular, round and anisotropic pointed quadratic space over F. Moreover, Q is wild and

Ξ := Θ_{d} = π^{− \frac{d}{2}} (1_{P} + j) ∈ V_{Q} (2)

is a unit of 𝔬_{Q} with

(3)  n_{Q}(Ξ) = β, t_{Q}(Ξ) = 2π^{− \frac{d}{2}} , V_{Q} = V_{P} ⊕ V_{P} Ξ,
(4)  λ_{Q}(u + vΞ) = min {λ_{P} (u), λ_{P} (v)}  (u, v ∈ V_{P}),
(5)  𝔬_{Q} = 𝔬_{P} ⊕ 𝔬_{P} Ξ, 𝔭_{Q} = 𝔭_{P} ⊕ 𝔭_{P} Ξ,
(6) e_{Q/F} = 1, Q̄ = ⟨⟨β̄⟩⟩ ⊗ P̄, texp(Q) = texp(P) − \frac{d}{2}.

Proof. The assertion μ ∈ 𝔬^{×} is trivial for d > 0 but holds also for d = 0 since in this case β̄ ∉ V 2 ; P̄ implies μ̄ = 1F̄ − β̄ ∉ V 2 ; P̄ by wildness of P. Next we show that Q is anisotropic. Otherwise, μ ∈ n_{P} (V_{P}) by Prop. 7.5 (a), and Lemma 8.13 yields an element w ∈ 𝔬_{P}^{×} with β = −n_{P} (w) + π^{−m}t_{P} (w), m = \frac{d}{2}, where the second summand belongs to 𝔭^{texp(P)−m} ⊆ 𝔭 by the hypothesis on d. Thus β̄ = nP̄(w̄) = w̄^{2}, a contradiction, and we have proved that Q is indeed anisotropic. Consulting (9.4.4- −6) for u = 0, v = 1_{P}, we end up with (3). Turning to (4), it suffices to show, by Lemma 7.12, that u, v ∈ 𝔬_{P}^{×} implies u + vΞ ∈ 𝔬_{Q}^{×}. Otherwise, observing (9.4.5),

λ_{Q}(u + vΞ) = \frac{1}{2}λ(n_{P} (u) + π^{−m}n_{P} (u, v) + βn_{P} (v))

were strictly positive, and since π^{−m}n_{P} (u, v) ∈ 𝔭 by Prop. 8.2 (c), we would again arrive at the contradiction β̄ ∈ V 2 ; P̄. Thus (4) holds, which directly implies (5), while (6) follows from (5) and (9.4.5,6). □

In 9.4, particularly (9.4.1), we are left with the case d = 2 texp(P), which turns out to be the most delicate. In order to get started, we require the following elementary but crucial observations.

9.8. Setting the stage for the case d = 2 texp(P). (a) Let K/k be a purely inseparable field extension of characteristic 2, exponent at most 1 and finite degree. Consider a scalar α ∈ k and a unital linear form s: K → k. We denote by by Q_{K;α,s} the pointed quadratic space over k with norm the Pfister quadratic form q_{K;α,s} of 3.4 and with base point 1_{K} ∈ K ⊆ K ⊕ Kj. Recall from Prop. 4.4 that Q_{K;α,s} = Q_{Cay(K;α,s)} is the pointed quadratic space corresponding to the flexible conic algebra Cay(K; α, s) that arises form K, α, s by means of the non-orthogonal Cayley-Dickson construction.

(b) Put m := texp(P) and let w_{0} be a normalized trace generator of P. Then w_{0} ∈ 𝔬_{P}^{×} by (8.3.1) and the map s_{w_{0}} : V_{P} → F, u ↦ π^{−m}n_{P} (u, w_{0}) is a unital linear form with s_{w_{0}}(𝔬_{P}) ⊆ 𝔬, s_{w_{0}}(𝔭_{P}) ⊆ 𝔭 by Prop. 8.2 (c) and since 𝔭_{P} = 𝔭𝔬_{P}. Thus we obtain an induced unital linear form

s̄_{w_{0}} : VP̄ ⟶ F̄, ū ⟼ s̄_{w_{0}}(ū) = \overline{π^{−m}n_{P} (u, w_{0})}, (1)

and given any α′ ∈ F̄, the notational conventions of (a) apply when VP̄/F̄ is viewed via 9.5 as a purely inseparable field extension of exponent at most 1.

9.9. Theorem. With the notations of 9.8, let

β ∈ 𝔬, μ := 1 − π^{2 texp(P)}β ∈ 𝔬^{×}, β_{0} := n_{P} (w_{0})β ∈ 𝔬, Q := ⟨⟨μ⟩⟩ ⊗ P. (1)

Then the following conditions are equivalent.

(i) Q is anisotropic and unramified.

(ii) Q is anisotropic and tame.

(iii) Q is anisotropic.

(iv) nexp_{P} (μ) = 2 texp(P). Moreover, if P is a pointed Pfister quadratic space, then these conditions are also equivalent to

(v) \overline{β_{0}} ∉ Im(℘_{VP̄,s̄_{w_{0}}} ).

Proof. In this section, we will not be able to give the proof in full but must restrict ourselves to showing the equivalence of (i)−(iv), relegating the rest to the next section. Since the implications (i) ⇒ (ii) ⇒ (iii) are obvious, it suffices to show (iv) ⇔ (iii) ⇒ (i).

(iv) ⟺ (iii). From 8.8 we deduce nexp_{P} (μ) ≥ 2 texp(P). Hence (iv) holds iff nexp_{P} (μ) ≤ 2 texp(P) iff Q is anisotropic by the local norm theorem 8.10 and Prop. 8.9 (a).

(iii) ⟹ (i). Setting d := 2 texp(P), we apply (9.4.5,6) for u = 0, v = w_{0} and obtain Ξ_{0} := w_{0}Θ_{d} ∈ 𝔬_{Q}^{×} and t_{Q}(Ξ_{0}) = 1. Thus Q is tame, and since P is wild, we conclude that P̄ is a proper pointed quadratic subspace of Q̄. Hence e_{Q/F} = 1, forcing Q to be unramified. □

Given β ∈ 𝔬, d ∈ ℤ with 0 ≤ d ≤ 2 texp(P), it follows from 8.8 that μ := 1−π^{d}β has norm exponent at least d, and if d is odd, Prop. 8.14 (a) yields a characterization in terms of β when equality holds. While a similar characterization for d = 2 texp(P) is presented in Thm. 9.9 (v) (though as yet unproved), we are now able to provide one for d even, d < 2 texp(P).

9.10. Corollary. Let d be an even integer such that 0 ≤ d < 2 texp(P).

(a) With β ∈ 𝔬 and μ := 1 − π^{d}β, the following conditions are equivalent. (i) μ ∈ 𝔬^{×} and nexp_{P} (μ) = d.

(ii) β̄ ∉ V 2 ; P̄.

(b) If d > 0, β_{i} ∈ 𝔬^{×} and μ_{i} := 1 − π^{d}β_{i} ∈ 𝔬^{×} for i = 1, 2, then

⟨⟨μ_{1}⟩⟩ ⊗ P ≅ ⟨⟨μ_{2}⟩⟩ ⊗ P ⟹ \overline{β_{1}} ≡ \overline{β_{2}} mod V 2 ; P̄.

Proof. (a) (i) ⟹ (ii). We have β ∈ 𝔬^{×} by Prop. 8.9 (b). For d = 0 the assertion follows from Prop. 8.9 (c). Now suppose d ≠ 0. Arguing indirectly, we assume that there exists an element w ∈ 𝔬_{P}^{×} with β̄ = w̄^{2} = \overline{n_{P} (w)}. Then n_{P} (w) = β + πβ′ for some β′ ∈ 𝔬, and Lemma 8.15 with m := \frac{d}{2} yields

μ = 1 − π^{d}β = (1 − π^{d+1}β′′)n_{P} (v′′)^{−1}

where, setting r := texp(P),

β′′ = −β′ + π^{r−m−1}α − π^{r+m−1}αβ + π^{2m−1}βn_{P} (w) ∈ 𝔬, v′′ = 1_{P} − π^{m}w ∈ 𝔬_{P}^{×}

since 0 < m < r. Now roundness of P and the definition of the norm exponent imply nexp_{P} (μ) ≥ d + 1, a contradiction.

(ii) ⟹ (i). By (9.7.1) we have μ ∈ 𝔬^{×} and Q is anisotropic by Prop. 9.7, forcing μ ∉ n_{P} (V_{P}^{×}) (Prop. 7.5 (a)) and d ≤ d′ := nexp_{P} (μ) ≤ 2 texp(P) (Thm. 8.10). Furthermore, by Prop. 8.9 (b), μ = μ′n_{P} (v′), μ′ = 1 − π^{d′}β′ for some β′ ∈ 𝔬^{×}, v′ ∈ 𝔬_{P}^{×}. In particular, μ′ ∈ 𝔬^{×} and Q ≅ Q′ := ⟨⟨μ′⟩⟩ ⊗ P. This implies e_{Q′/F} = e_{Q/F} = 1 by (9.7.6), so d′ < 2 texp(P) (Thm. 9.9) is even (Prop. 9.6), and we are allowed to apply Prop. 9.7 to Q′ by the implication (i) ⇒ (ii) already established. Thus (9.7.6) yields d = d′.

(b) We put Q_{i} := ⟨⟨μ_{i}⟩⟩ ⊗ P for i = 1, 2. If V 2 ; P̄ contains \overline{β_{1}} but not \overline{β_{2}}, then μ_{1}, μ_{2} have different norm exponents by (a), so Q_{1}, Q_{2} cannot be isomorphic (Props. 8.9 (d), 7.5 (b)). We may therefore assume \overline{β_{1}}, \overline{β_{2}} ∉ V 2 ; P̄. As in the proof of Cor. 8.14 we have μ := μ_{1}μ_{2} = 1 − π^{d}β, β := β_{1} + β_{2} − π^{d}β_{1}β_{2} ∈ 𝔬. Assuming β̄ = \overline{β_{1}} − \overline{β_{2}} ∉ V 2 ; P̄ would force ⟨⟨μ⟩⟩ ⊗ P to be anisotropic by Prop. 9.7, hence μ_{1}, μ_{2} to fall into distinct norm classes relative to P. But then Q_{1}, Q_{2} would not be isomorphic, a contradiction. □

9.11. Examples. (a) Let m be an integer with 0 ≤ m < texp(P) and suppose we are given an element γ ∈ 𝔬^{×} such that γ̄ ∈ V 2 ; P̄. If μ := 1 − π^{2m}γ is a unit in 𝔬 (automatic unless m = 0), Cor. 9.10 (a) implies nexp_{P} (μ) > 2m, so it is a natural question to ask whether a more precise estimate for the norm exponent of μ can be given. Unless specific properties of γ are taken into account, the answer is no. To see this, let d ∈ ℤ with d > 2m, β ∈ 𝔬, and w ∈ 𝔬_{P}^{×} with w̄ ≠ 1P̄. Then Lemma 8.15 implies

1 − π^{2m}γ ≡ 1 − π^{d}β mod n_{P} (P^{×}), γ̄ = w̄^{2}

for

γ = −n_{P} (w) + π^{d−2m}β + π^{texp(P)−m}α − π^{texp(P)+d−m}αβ + π^{d}βn_{P} (w) ∈ 𝔬^{×}.

Hence nexp_{P} (1 − π^{2m}γ) = ∞ for d > 2 texp(P) by the local norm theorem 8.10, and, by Cors. 8.14,9.10, β may be so chosen that nexp_{P} (1−π^{2m}γ) attains any finite pre-assigned value d with 2m < d < 2 texp(P) provided V 2 ; P̄ ≠ F̄.

(b) Let d ∈ ℤ, 0 < d ≤ 2 texp(P), β, β′ ∈ 𝔬^{×} and put μ := 1 − π^{d}β, μ′ = 1 − π^{d}β′ ∈ 𝔬^{×}. We wish to refute the converse of Cors. 8.14 (b) and 9.10 (b) by showing that β̄ = β̄′ does not imply μ ≡ μ′ mod n_{P} (V_{P}^{×}). Indeed, if β ≠ β′, then β̄ = β̄′ amounts to the same as μ′ = 1−π^{d}β −π^{q}γ for some integer q > d and some γ ∈ 𝔬^{×}, allowing us to conclude

μμ′ = μ^{2}(1 − π^{q}μ^{−1}γ) ≡ 1 − π^{q}μ^{−1}γ mod n_{P} (V_{P}^{×}).

Therefore,

• μ ≡ μ′ mod n_{P} (V_{P}^{×}) for q > 2 texp(P) (Thm. 8.10), but

• μ ≢ μ′ mod n_{P} (V_{P}^{×}) for d < q ≤ 2 texp(P), provided γ̄ ∉ V 2 ; P̄ if q < 2 texp(P) is even (Cors. 8.14,9.10), and w̄_{0}^{2}γ̄ ∉ Im(℘_{VP̄;s̄_{w_{0}}} ) if

d = 2 texp(P) (Thm. 9.9). □

The preceding results on the behavior of the ramification index, the pointed quadratic residue space and the trace exponent under the passage from P to ⟨⟨μ⟩⟩ ⊗ P, μ ∈ F^{×}, can be stated in a particularly concise way when addressed to pointed Pfister quadratic spaces by combining them with the embedding property of Prop. 7.6. In order to do so, we introduce the following terminology.

9.12. Scalars of standard type. We say a scalar μ ∈ F has standard type relative (or with respect) to P if it satisfies one of the following mutually exclusive conditions.

(a) μ is a prime element of 𝔬 (possibly distinct from π).

(b) μ = 1 − π^{d}β for some odd integer d with 0 ≤ d < 2 texp(B) and some β ∈ 𝔬^{×}.

(c) μ = 1 − π^{d}β for some even integer d with 0 ≤ d < 2 texp(B) and some

β ∈ 𝔬 with β̄ ∉ V 2 ; P̄.

9.13. Theorem. Let P be a pointed n-Pfister quadratic space over F that is anisotropic and wild of ramification index e_{P/F} = 1. For Q to be a wild anisotropic pointed (n + 1)-Pfister quadratic space over F into which P embeds as a pointed quadratic subspace it is necessary and sufficient that Q be a pointed quadratic space isomorphic to ⟨⟨μ⟩⟩ ⊗ P, for some scalar μ ∈ F of standard type relative to P. In this case, precisely one of the following implications holds.

(a) If μ is a prime element in 𝔬, then

e_{Q/F} = 2, Q̄ ≅ P̄, texp(Q) = texp(P).

(b) If μ = 1 − π^{d}β for some odd integer d with 0 ≤ d < 2 texp(P) and some β ∈ 𝔬^{×}, then

e_{Q/F} = 2, Q̄ ≅ P̄, texp(Q) = texp(P) − \frac{d − 1}{2}.

(c) If μ = 1 − π^{d}β for some even integer d, 0 ≤ d < 2 texp(P) and some β ∈ 𝔬 with β̄ ∉ V 2 ; P̄, then

e_{Q/F} = 1, Q̄ ≅ ⟨⟨β̄⟩⟩ ⊗ P̄, texp(Q) = texp(P) − \frac{d}{2}.

Proof. By Props. 9.2,9.6,9.7, the condition is sufficient, and (a)−(c) hold. Conversely, suppose Q is a pointed anisotropic wild (n+1)-Pfister quadratic space over F containing P as a pointed quadratic subspace. Up to isomorphism, Q = ⟨⟨μ⟩⟩⊗P for some μ ∈ F^{×} by the embedding property (Prop. 7.6), where the reduction of 9.1 (b) allows us to assume that μ ∈ 𝔬^{×} is a unit in 𝔬. Q being anisotropic implies 0 ≤ d := nexp_{P} (μ) ≤ 2 texp(P) by the local norm theorem 8.10 and without loss μ = 1 − π^{d}β for some β ∈ 𝔬. Since Q is wild, we conclude d < 2 texp(P) from the part of Thm. 9.9 already established, and if d is odd, then β ∈ 𝔬^{×} (Prop. 8.14 (a)), while if d is even, then β̄ ∉ V 2 ; P̄ (Cor. 9.10). In any event, μ has standard type relative to P. □

9.14. Corollary. With P as in Thm. 9.13, let Q be a pointed quadratic space over F.

(a) The following conditions are equivalent.

(i) Q is an anisotropic pointed (n + 1)-Pfister quadratic space into which P embeds as a pointed quadratic subspace such that e_{Q/F} = 2 and texp(Q) = texp(P).

(ii) Q ≅ ⟨⟨πβ⟩⟩ ⊗ P or Q ≅ ⟨⟨1 − πβ⟩⟩ ⊗ P for some β ∈ 𝔬^{×}.

In this case Q̄ ≅ P̄.

(b) The following conditions are equivalent.

(i) Q is an anisotropic pointed (n + 1)-Pfister quadratic space into which P embeds as a pointed quadratic subspace such such that e_{Q/F} = 1 and texp(Q) = texp(P).

(ii) Q ≅ ⟨⟨μ⟩⟩ ⊗ P for some μ ∈ 𝔬 with μ̄ ∉ V 2 ; P̄.

In this case, Q̄ ≅ ⟨⟨μ̄⟩⟩ ⊗ P.

Proof. In (a) and (b), condition (i) implies that Q is wild (Prop. 8.2 (a)). Hence the assertions follow immediately from Thm. 9.13. □

There is an analogue of Thm. 9.13 dealing with tame rather than wild enlargements of pointed n-Pfister quadratic spaces. We omit the proof since it proceeds along the same lines as the one of Thm. 9.13, applying Thm. 9.9 in full rather than Props. 9.2,9.6,9.7.

9.15. Theorem. Keeping the notations of 9.8 (b), let P be a pointed n-Pfister quadratic space over F that is anisotropic and wild of ramification index e_{P/F} = 1. For Q to be a tame and anisotropic (n + 1)-Pfister pointed quadratic space over F into which P embeds as a pointed quadratic subspace it is necessary and sufficient that Q be a pointed quadratic F -space and there exist an element β ∈ 𝔬 with

Q ≅ ⟨⟨μ⟩⟩ ⊗ P, μ := 1 − π^{2 texp(B)}β, \overline{β_{0}} ∉ Im(℘_{VP̄,s̄_{w_{0}}} ), β_{0} := n_{P} (w_{0})β.

□

10. λ-normed and λ-valued conic algebras.

In order to illuminate the intuitive background of the present section, we recall the notion of an absolute-valued algebra. Following Albert [1] (see also Palacios [37]), an absolute-valued algebra is a non-associative real algebra A equipped with a norm x ↦ ∥x∥ that permits composition: ∥xy∥ = ∥x∥ ∥y∥ for all x, y ∈ A. Since absolute valued algebras obviously have no zero divisors, the finite-dimensional ones are division algebras, hence exist only in dimensions 1, 2, 4, 8 (Albert [1] gave an adhoc proof of this result, the Bott-Kervaire-Milnor thoerem not having been known at the time). By contrast, natural analogues of absolute-valued algebras over 2-Henselian fields will be discussed in the present section that exist in all dimensions 2^{n}, n = 0, 1, 2 ….

Throughout we continue to work over a fixed 2-Henselian field F as in 7.7 and alert the reader to the terminological conventions of 7.14. All vector spaces, algebras, etc. over F are tacitly assumed to be finite-dimensional.

10.1. The basic concepts. A conic algebra C over F is said to be λ-normed if the following conditions hold.

(i) C is non-singular, round and anisotropic.

(ii) λ_{C} is sub-multiplicative: λ_{C}(xy) ≥ λ_{C}(x) + λ_{C}(y) for all x, y ∈ C. We speak of a λ-valued conic algebra if C satisfies (i), and if instead of (ii) the following stronger condition holds:

(iii) λ_{C} is multiplicative: λ_{C}(xy) = λ_{C}(x) + λ_{C}(y) for all x, y ∈ C. The norm of a λ-valued conic algebra C over F will typically not permit composition (for example, if the dimension of C differs from 1, 2, 4, 8) but, remarkably, its failure to do so is not detected by λ since λ_{C} being multiplicative by (7.14.1) amounts to λ(n_{C}(xy)) = λ(n_{C}(x)n_{C}(y)) for all x, y ∈ C. This looks like a pretty far-fetched phenomenon but, in fact, turns out to be quite common.

We start with a trivial but useful observation.

10.2. Proposition. (a) λ-valued conic algebras over F are division algebras.

(b) A composition algebra over F is a λ-valued conic algebra if and only if it is a division algebra. □

10.3. Proposition. Let C be a non-singular, round and anisotropic conic algebra over F. Then C is λ-normed if and only if 𝔬_{C} ⊆ C is an 𝔬-subalgebra and 𝔭_{C} ⊆ 𝔬_{C} is an ideal with 𝔭_{C}^{2} ⊆ 𝔭𝔬_{C}. In this case, C̄ := 𝔬_{C}/𝔭_{C} is a conic algebra over F̄ whose norm, trace and conjugation are given by the formulas

(1)  nC̄(x̄) = \overline{n_{C}(x)},
(2)  tC̄(x̄) = \overline{t_{C}(x)},
(3)  \binom{}{x̄}^{∗} = \overline{(x^{∗})}

for all x ∈ 𝔬_{C}. Moreover, the norm of C̄ is round and anisotropic.

Proof. By Lemma 7.12, sub-multiplicativity of λ_{C} amounts to

λ_{C}(xy) ≥ λ_{C}(x) + λ_{C}(y)  (x, y ∈ C, 0 ≤ λ_{C}(x), λ_{C}(y) ≤ \frac{1}{2}).

The first part of the proposition follows from this at once. The second part is a restatement of Prop. 7.10. □

Remark. The conic algebra C̄ = 𝔬_{C}/𝔭_{C} is called the residue algebra of C. If C is wild, we do not know whether this residue algebra always agrees with the purely inseparable extension field of F̄ attached to C via 9.5, though it does if C is a composition algebra [41, Prop. 1].

10.4. Corollary. With the notations of Prop. 10.3 suppose in addition that C has ramification index e_{C/F} = 1. Then:

(a) C is λ-normed if and only if 𝔬_{C} ⊆ C is an 𝔬-subalgebra.

(b) C is λ-valued if and only if C is λ-normed and C̄ is a division algebra.

Proof. (a) 𝔭_{C} = 𝔭𝔬_{C}.

(b) Consulting Lemma 7.12 again, λ_{C} is multiplicative iff λ_{C}(xy) = 0 for all x, y ∈ 𝔬_{C}^{×} iff C is λ-normed and C̄ is a division algebra. □

We now proceed to re-examine the main results of the preceding section within the framework of λ-normed and λ-valued conic algebras.

10.5. Convention. For the remainder of this section, we fix a λ-normed conic algebra B over F having ramification index e_{B/F} = 1.

10.6. Proposition. If μ is a prime element in 𝔬, then C := Cay(B, μ) is a λnormed conic algebra over F with C̄ = B̄ as conic F̄-algebras. Moreover, C is λ-valued if and only if B is λ-valued.

Proof. By Prop. 9.2, C is non-singular, round and anisotropic. Combining Prop. 10.3 with (1.10.1), we conclude that C is λ-normed. It remains to show that if B is λ-valued, so is C. By Lemma 7.12, we must show λ_{C}(x_{1}x_{2}) = λ_{C}(x_{1})+λ_{C}(x_{2}) for all x_{i} = u_{i} + v_{i}j ∈ C, u_{i}, v_{i} ∈ B, 0 ≤ λ_{C}(x_{i}) ≤ \frac{1}{2}, i = 1, 2. There are four cases: (i) λ_{C}(x_{1}) = λ_{C}(x_{2}) = 0, (ii) λ_{C}(x_{1}) = 0, λ_{C}(x_{2}) = \frac{1}{2}, (iii) λ_{C}(x_{1}) = \frac{1}{2}, λ_{C}(x_{2}) = 0, (iv) λ_{C}(x_{1}) = λ_{C}(x_{2}) = \frac{1}{2}. We only treat (iv) and leave the other three cases to the reader. From (9.2.1) we deduce u_{i} ∈ 𝔭_{B}, v_{i} ∈ 𝔬_{B}^{×} for i = 1, 2 and (1.10.1) yields

x_{1}x_{2} = u + vj, u = u_{1}u_{2} + μv_{2}^{∗}v_{1}, v = v_{2}u_{1} + v_{1}u_{2}^{∗},

where λ_{B}(u) = 1, λ_{B}(v) ≥ 1, hence λ_{C}(x_{1}x_{2}) = 1 = λ_{C}(x_{1}) + λ_{C}(x_{2}) □

10.7. Example. Specializing Prop. 10.6 to (iterated) Laurent series fields of characteristic not 2, we recover examples of λ-valued conic algebras that originally go back to to Brown [7, pp. 421-422]. In a slightly more general vein, let k be any field, L/k a separable quadratic field extension and write

A = Cay(L; μ_{1}, …, μ_{n−1})  (n ∈ ℤ, n ≥ 1)

for the k-algebra arising from L and scalars μ_{1}, …, μ_{n−1} ∈ k^{×} by means of the Cayley-Dickson process as in 1.12. Then A is a flexible conic algebra with norm an n-Pfister quadratic form. We now assume that A is a division algebra, forcing A to be non-singular, round and anisotropic. Consider the field F = k((t)) of formal Laurent series in a variable t with coefficients in k, which is complete and therefore Henselian under the standard discrete valuation λ: F → ℤ_{∞}. Setting

B := A ⊗_{k} F = A((t)),

we obtain a flexible conic division F -algebra whose norm is an anisotropic n-Pfister quadratic form over F. Using (7.9.1), a straightforward verification shows

λ_{B}( ∑_{r≫−∞}^{∞} a_{r}t^{r}) = min {r ∈ ℤ | a_{r} ≠ 0}  (a_{r} ∈ A, r ∈ ℤ),

which immediately implies that B is an unramified λ-valued conic algebra over F. By Prop. 10.6 we thus find in C := Cay(B, t) a λ-valued conic algebra over F having dimension 2^{n+1}. Starting from A = L (i.e., from n = 1) and continuing in this way, we obtain λ-valued conic algebras over appropriate iterated Laurent series fields in all dimensions 2^{n}, n = 0, 1, 2, . . ..

10.8. Example. We specify Example 10.7 a bit further by setting k = ℝ, L = ℂ, n = 3, μ_{1} = μ_{2} = −1. Then A = 𝕆, the real algebra of Graves-Cayley octonions, and B = 𝕆((t)) is the unique unramified octonion division algebra over F = ℝ((t)). Moreover, the 16-dimensional conic division algebra

C = Cay(B, t) = Cay(F; −1, −1, −1, t)

over F contains B′ := Cay(F; −1, −1, t) as a ramified octonion subalgebra. In particular, B and B′ are not isomorphic, allowing us to conclude from [7, Thm. 2] that the subalgebra B′ ⊆ C does not satisfy the embedding property 1.11.

10.9. Proposition. If B as in 10.5 is tame and μ ∈ 𝔬^{×} \ n_{B}(B^{×}), then C := Cay(B, μ) is a λ-normed conic algebra over F with C̄ = Cay(B̄, μ̄) as conic F̄- algebras. Moreover, C is λ-valued if and only if C̄ is a division algebra.

Proof. By Prop. 9.3, C is a tame non-singular, round and anisotropic conic algebra over F with e_{C/F} = 1. Moreover, 𝔬_{B} being a 𝔬-subalgebra of B by Prop. 10.3, we conclude from (9.3.2) that 𝔬_{C} is an 𝔬-subalgebra of C. Now everything follows from Cor. 10.4. □

Dealing with the case that B as in 10.5 is wild turns out to be more troublesome. We not only need a few preparations but also have to add an extra hypothesis by requiring that the conic algebras involved be flexible.

10.10. Proposition. Let C be a flexible λ-normed conic algebra over F. Then

(1)  λ(t_{C}(x)) ≥ ω(C) + λ_{C}(x),
(2)  λ_{C}([x_{1}, x_{2}]) ≥ ω(C) + λ_{C}(x_{1}) + λ_{C}(x_{2}),
(3)  λ_{C}(x − x^{∗}) ≥ ω(C) + λ_{C}(x)

for all x, x_{1}, x_{2} ∈ C^{×}.

Proof. (1) follows immediately from (8.4.1). To establish (2), we combine (1) with (2.4.2),(7.9.5),(8.7.2) and use the fact that λ_{C} is sub-multiplicative. Finally, applying (1) and (2.1.6), we obtain

λ_{C}(x − x^{∗}) = λ_{C}(2x − t_{C}(x)1_{C}) ≥ min {e_{F} + λ_{C}(x), ω(C) + λ_{C}(x)},

and (3) follows from (8.7.2). □

10.11. Lemma. Suppose B as in 10.5 is wild and μ ∈ 𝔬^{×} has the form

μ = 1 − π^{d}β, d ∈ ℤ, 0 ≤ d ≤ 2 texp(B), β ∈ 𝔬^{×}. (1)

Setting C = Cay(B, μ) and

0 ≤ m := ⌊\frac{d}{2}⌋ < texp(B), Θ_{d} := π^{−m}(1_{B} + j) ∈ C (2)

as in (9.4.1), the relations

(3) u(vΘ_{d}) = π^{−m}[u, v] + (vu)Θ_{d},
(4) (vΘ_{d})u = π^{−m}v(u − u^{∗}) + (vu^{∗})Θ_{d},

(v_{1}Θ_{d})(v_{2}Θ_{d}) = (π^{−2m}[v_{1}, v_{2} − v_{2}^{∗}] − π^{d−2m}βv_{2}^{∗}v_{1}) + π^{−m}(t_{B}(v_{2})v_{1} − [v_{1}, v_{2}])Θ_{d} (5)

hold for all u, v, v_{1}, v_{2} ∈ B. Moreover, if B is flexible, then 𝔒 := 𝔬_{B} ⊕ 𝔬_{B}Θ_{d} is an 𝔬-subalgebra of C.

Proof. A slightly involved but straightforward computation using the transition formulas

u + vΘ_{d} = (u + π^{−m}v) + π^{−m}vj, u + vj = (u − v) + π^{m}vΘ_{d}

implies (3)−(5). Combining these with Prop. 10.10 leads to the final assertion of the lemma. □

10.12. Theorem. Suppose B as in 10.5 is flexible and wild,

μ = 1 − π^{d}β, d ∈ ℤ, 0 ≤ d < 2 texp(B), β ∈ 𝔬^{×},

and d is odd. Then C := Cay(B, μ) is a flexible λ-normed conic algebra over F with C̄ = B̄ as conic F̄-algebras. Moreover, C is λ-valued if and only if B is λ-valued.

Proof. The proof is similar to, but a bit more complicated than, the one of Prop. 10.6. By Prop. 9.6, C is non-singular, round and anisotropic. Combining the final statement of Lemma 10.11 with (9.6.4), we conclude that 𝔬_{C} ⊆ C is an 𝔬-subalgebra which, thanks to (10.11.3−5) and to d being odd contains 𝔭_{C} as an ideal with 𝔭_{C}^{2} ⊆ 𝔭𝔬_{C}. Thus C is λ-normed (Prop. 10.3), and it remains to show that if B is λ-valued, so is C. Let x_{i} = u_{i} + v_{i}Π ∈ C^{×}, u_{i}, v_{i} ∈ B, i = 1, 2 and x_{1}x_{2} = u + vΠ, where (10.11.3−5) imply

(1) u = u_{1}u_{2} + π^{−m}([u_{1}, v_{2}] + v_{1}(u_{2} − u_{2}^{∗})) + π^{−2m}[v_{1}, v_{2} − v_{2}^{∗}] − πβv_{2}^{∗}v_{1},
(2) v = v_{2}u_{1} + v_{1}u_{2}^{∗} + π^{−m}(t_{B}(v_{2})v_{1} − [v_{1}, v_{2}]).

We must show λ_{C}(x_{1}x_{2}) = λ_{C}(x_{1})+λ_{C}(x_{2}). To this end, invoking Lemma 7.12, we may assume 0 ≤ λ_{C}(x_{i}) ≤ \frac{1}{2}, i = 1, 2. Since conjugation is an algebra involution of C leaving λ_{C} invariant, there are three cases: (i) λ_{C}(x_{1}) = λ_{C}(x_{2}) = 0, (ii) λ_{C}(x_{1}) = 0, λ_{C}(x_{2}) = \frac{1}{2}, (iii) λ_{C}(x_{1}) = λ_{C}(x_{2}) = \frac{1}{2}. Among these cases, we treat only (iii) since the other ones can be treated analogously. In (iii) we have u_{1}, u_{2} ∈ 𝔭_{B}, v_{1}, v_{2} ∈ 𝔬_{B}^{×}, observe Prop. 10.10 and obtain u ≡ −πβv_{2}^{∗}v_{1} mod 𝔭_{B}^{2} by (1), hence λ_{B}(u) = 1, while (2) yields λ_{B}(v) ≥ 1. Therefore λ_{C}(x_{1}x_{2}) = 1 = λ_{C}(x_{1})+λ_{C}(x_{2}). □

10.13. Theorem. Suppose B as in 10.5 is flexible and wild,

μ = 1 − π^{d}β, d ∈ ℤ, 0 ≤ d < 2 texp(B), β ∈ 𝔬, β̄ ∉ V 2 ; B̄ (1)

and d is even. Then C := Cay(B, μ) is a wild λ-normed conic algebra over F with C̄ ≅ Cay(B̄, β̄) as conic F̄-algebras. Moreover, C is λ-valued if and only if C̄ is a division algebra.

Proof. By Prop. 9.7, C is a non-singular, round and anisotropic conic F -algebra. Moreover, C is wild of ramification index 1. The final statement of Lemma 10.11 combined with (9.7.5) shows that 𝔬_{C} ⊆ C is an 𝔬-subalgebra, forcing C to be λnormed (Cor. 10.4 (a)). Moreover, writing Cay(B̄, β̄) = B̄ ⊕ B̄j′, j^{′2} = β̄1B̄ as in 1.10, consulting (10.11.3−5) and observing d < texp(B) = ω(B), Prop. 10.10 shows that (in the notations of Prop. 9.7) the assignment u + vΞ ↦ ū + v̄j′ determines an isomorphism C̄ →^{∼} Cay(B̄, β̄) of F̄-algebras. The final statement of the theorem follows immediately from Cor. 10.4 (b). □

10.14. Corollary. With the notations and assumptions of Thm. 10.13, suppose in addition that B̄/F̄ is a purely inseparable field extension of exponent at most 1. Then C is a λ-valued conic algebra over F with C̄ ≅ B̄(\sqrt{β̄}).

Proof. It suffices to note that the last condition of (10.13.1) makes Cay(B̄, β̄) = B̄(\sqrt{β̄}) (cf. 1.12, Case 2) a division algebra. □

Remark. The additional hypothesis in Cor. 10.14 is fulfilled if, e.g., B is a composition algebra (Remark to 10.3).

10.15. Examples. In Brown’s examples of conic division algebras (cf. 10.7), one basically keeps building up ramified λ-valued conic algebras over iterated Laurent series fields of characteristic not 2. By contrast, we will now be able to construct wild λ-valued conic algebras of ramification index 1 over appropriate Henselian fields of characteristic zero. Let k be any field of characteristic 2 and write K for the field of rational functions in an infinite number of variables over k. Then [K : K^{2}] = ∞. Pick an infinite chain

K = K_{0} ⊂ K_{1} ⊂ ⋯ ⊂ K_{n−1} ⊂ K_{n} ⊂ ⋯

of purely inseparable field extensions of K having exponent at most 1 and [K_{n} : K] = 2^{n} for all integers n ≥ 0. Following Teichmüller [49], there is an essentially unique complete field F under a discrete valuation λ: F → ℤ_{∞} such that F has characteristic zero, residue field F̄ = K and absolute ramification index e_{F} = 1. For n ≥ 1 choose β_{n} ∈ 𝔬^{×} such that K_{n} = K_{n−1}(\sqrt{β̄_{n}}), put μ_{n} = 1 − β_{n}, observe (8.7.1) and apply Cor. 10.14 successively for d = 0 and n = 1, 2, 3, … to conclude that the Cayley-Dickson process leads to a wild λ-valued conic algebra

C_{n} := Cay(F; μ_{1}, …, μ_{n})

over F having dimension 2^{n} and ramification index 1 such that C̄_{n} ≅ K_{n}.

10.16. Corollary. Let Q be a pointed Pfister quadratic space over F that is anisotropic and wild of ramification index e_{Q/F} = 1. Then there exists a flexible λ-valued conic algebra C over F such that Q_{C} ≅ Q and C̄/F̄ is a purely inseparable field extension of exponent at most 1.

Proof. Arguing by induction, we let Q be a pointed (n + 1)-Pfister quadratic space and pick a pointed n-Pfister quadratic subspace P ⊆ Q. Clearly, P is wild with e_{P/F} = 1. By Theorem 9.13, some scalar μ of standard type relative to P satisfies Q = ⟨⟨μ⟩⟩⊗P up to isomorphism, and since the implications (a),(b) of that theorem do not hold for μ, implication (c) does. On the other hand, the induction hypothesis leads to a flexible λ-valued conic algebra B over F with Q_{B} ≅ P such that B̄/F̄ is a purely inseparable field extension of exponent at most 1. By Cor. 10.14, C := Cay(B, μ) is a flexible λ-valued conic algebra over F with Q_{C} = ⟨⟨μ⟩⟩ ⊗ P = Q and C̄ ≅ B̄(\sqrt{β̄}). □

We still haven’t closed the gap in our proof of Thm. 9.9 but will now be able to do so by appealing to the connection with conic algebras. In view of Cors. 10.14,10.16, the missing equivalence of (v) and (i)−(iv) in Thm. 9.9 will be a consequence of the following result.

10.17. Theorem. Suppose B as in 10.5 is flexible, wild and λ-normed having B̄/F̄ as a purely inseparable field extension of exponent at most 1 and, with the notations of 9.8 (b), let

β ∈ 𝔬, μ := 1 − π^{2 texp(B)}β ∈ 𝔬^{×}, β_{0} := n_{B}(w_{0})β. (1)

Then C := Cay(B, μ) is anisotropic if and only if \overline{β_{0}} ∉ Im(℘_{B̄,s̄_{w_{0}}} ). In this case, setting

Ξ_{0} := π^{− texp(B)}(w_{0} + w_{0}j) ∈ C, (2)

we obtain the relations

(3)  t_{C}(Ξ_{0}) = 1, n_{C}(Ξ_{0}) = β_{0}, C = B ⊕ BΞ_{0},
(4)  λ_{C}(u + vΞ_{0}) = min {λ_{B}(u), λ_{B}(v)}  (u, v ∈ B),
(5)  𝔬_{C} = 𝔬_{B} ⊕ 𝔬_{B}Ξ_{0}, 𝔭_{C} = 𝔭_{B} ⊕ 𝔭_{B}Ξ_{0}

and C is λ-normed with QC̄ ≅ Q_{B̄;β̄_{0},s̄_{w_{0}}} .

Proof. As in the proof of the implication (iii) ⇒ (i) in Thm. 9.9, we put d := 2 texp(B), Ξ := Θ_{d} and have Ξ_{0} = w_{0}Ξ. Thus the first two relations of (3) follow from (9.4.5,6), while the last one is a straightforward consequence of the fact that B is λ-valued by Cor. 10.4 (b), hence a division algebra (Prop. 10.2 (a)). Setting m := texp(B), we let v ∈ B and compute, using (1.10.1,2),(2.4.1) and (1),

n_{C}(vΞ_{0}) = π^{−2m}n_{C}(v(w_{0} + w_{0}j)) = π^{−2m}n_{C}(vw_{0} + (w_{0}v)j)
= π^{−2m}(n_{B}(vw_{0}) − μn_{B}(w_{0}v)) = π^{−2m}n_{B}(vw_{0})(1 − μ) = n_{B}(vw_{0})β.

But since B̄/F̄ is a purely inseparable field extension of exponent at most 1, we have n_{B}(vw_{0}) ≡ n_{B}(v)n_{B}(w_{0}) mod 𝔭 and conclude

n_{C}(vΞ_{0}) ≡ n_{B}(v)β_{0} mod 𝔭. (6)

Suppose first that C is anisotropic. Then C̄ is an anisotropic conic algebra over F̄ containing B̄ as a subalgebra and l := \overline{Ξ_{0}} ∈ C̄ as a distinguished element with tC̄(l) = 1, nC̄(l) = \overline{β_{0}}, nC̄(ū, l) = s̄_{w_{0}}(ū) for all u ∈ 𝔬_{B}. Moreover, (6) implies nC̄(v̄l) = \overline{β_{0}}v̄^{2} for all v ∈ 𝔬_{B}. Writing B̄ ⊕ B̄j′ for the vector space underlying the pointed quadratic space Q_{B̄;β̄_{0},s̄_{w_{0}}} , the assignment ū + v̄j′ ↦ ū + v̄l therefore and by (2.2.2) gives an embedding φ from Q_{B̄;β̄_{0},s̄_{w_{0}}} to QC̄ of pointed quadratic spaces. Comparing dimensions (observe e_{C/F} = 1 by (i) of Thm. 9.9), φ: Q_{B̄;β̄_{0},s̄_{w_{0}}} →^{∼} QC̄ is, in fact, an isomorphism, and since QC̄ is anisotropic, so is Q_{B̄;β̄_{0},s̄_{w_{0}}} . Now \overline{β_{0}} ∉ Im(℘_{B̄,s̄_{w_{0}}} ) follows from Cor. 3.10 (a). Moreover, we claim that (4) holds (which immediately implies (5)). As usual, we may assume u, v ∈ 𝔬_{B}^{×}, which yields λ_{C}(u+vΞ_{0}) ≥ 0, and if this were strictly positive, we would end up with φ(ū+v̄j′) = ū + v̄l = 0, forcing ū = v̄ = 0, a contradiction. Suppose next \overline{β_{0}} ∉ Im(℘_{B̄,s̄_{w_{0}}} ) and consider the full 𝔬-lattice

𝔒 := 𝔬_{B} ⊕ 𝔬_{B}Ξ_{0} ⊆ C, (7) on which n_{C} takes integral values. More precisely, (6) and (2.2.2) imply

n_{C}(u + vΞ_{0}) ≡ n_{B}(u) + π^{−m}n_{B}(v^{∗}u, Ξ_{0}) + n_{B}(v)β_{0} mod 𝔭 (8)

for all u, v ∈ 𝔬_{B}. Reducing mod 𝔭, we obtain a pointed quadratic space

𝔒̄ := 𝔒 ⊗_{𝔬} F̄ = B̄ ⊕ B̄l′, l′ := Ξ_{0} ⊗_{𝔬} 1F̄,

over F̄, and the assignment ū ⊕ v̄j′ ↦ ū + v̄l′ by (7),(8) gives an isomorphism from Q_{B̄;β̄_{0},s̄_{w_{0}}} onto 𝔒̄. The former being anisotropic by Cor. 3.10 (a), so is the latter. But then C must be anisotropic as well since every non-zero element x ∈ C satisfies π^{m}x ∈ 𝔒 \ 𝔭𝔒 for some integer m.

It remains to show that C is λ-normed provided it is anisotropic. By Cor. 10.4 (a), it suffices to show that 𝔬_{C} ⊆ C is an 𝔬-subalgebra. In order to do so, we use Lemma 10.11 to derive the following formulas by a straightforward computation, for all u, v, v_{1}, v_{2} ∈ B.

(9)  vΞ_{0} = π^{−m}[v, w_{0}] + (w_{0}v)Ξ,
(10)  vΞ = π^{−m}[w_{0}, L_{w_{0}}^{−1}v] + (L_{w_{0}}^{−1}v)Ξ_{0},
(11)  u(vΞ_{0}) = π^{−m}(u[v, w_{0}] + [u, w_{0}v]) + ((w_{0}v)u)Ξ,
(12)  (vΞ_{0})u = π^{−m}([v, w_{0}]u + (w_{0}v)(u − u^{∗})) + ((w_{0}v)u^{∗})Ξ
(13) (v_{1}Ξ_{0})(v_{2}Ξ_{0}) = π^{−2m}([v_{1}, w_{0}][v_{2}, w_{0}] + [[v_{1}, w_{0}], w_{0}v_{2}]+
[w_{0}v_{1}, w_{0}v_{2} − (w_{0}v_{2})^{∗}] + (w_{0}v_{1})([v_{2}, w_{0}] − [v_{2}, w_{0}]^{∗}))−

β(w_{0}v_{2}^{∗})(w_{0}v_{1}) + π^{−m}((w_{0}v_{2})[v_{1}, w_{0}] + (w_{0}v_{1})[v_{2}, w_{0}]^{∗}+

t_{B}(w_{0}v_{2})w_{0}v_{1} − [w_{0}v_{1}, w_{0}v_{2}])Ξ.

Since B is λ-valued, (10) implies 𝔬_{B}Ξ ⊆ 𝔬_{C}, and then (11)−(13) combine with Prop. 10.10 to establish 𝔬_{C} as an 𝔬-subalgebra of C. □

10.18. Corollary. Suppose in Thm. 10.17 that B is an associative composition division algebra and \overline{β_{0}} ∉ Im(℘_{B̄;β̄_{0},s̄_{w_{0}}} ). Then C is an unramified composition division algebra over F with C̄ ≅ Cay(B̄; \overline{β_{0}}, s̄_{w_{0}}) as a non-orthogonal Cayley-Dickson construction.

Proof. Composition algebras are classified by their norms. □

Remark. If C as in Thm. 10.17 is anisotropic, its pointed quadratic residue space is described explicitly by the theorem. But C is also a λ-normed conic algebra, making C̄ canonically a conic algebra (over F̄) in its own right. It would be interesting to obtain an equally explicit description of that algebra. Cor. 10.18 provides one if C is a composition algebra but it is not at all clear whether this description prevails in the general case, nor whether C is always a λ-valued conic algebra.

11. Applications to composition algebras.

There are obvious and less obvious applications of the preceding results to composition algebras. Working over a fixed 2-Henselian field F of arbitrary characteristic as before (cf. 7.7), the obvious ones may be described as follows.

11.1. Translations. Since composition division algebras over F are classified by their norms and are λ-valued conic F -algebras by Prop. 10.2 (b), the results of Sections 8, 9 translate immediately into this more special setting, where the ones in Section 9 in particular yield explicit descriptions of how the valuation data ramification index, residue algebra and trace exponent behave under the Cayley-Dickson construction. Rather than carrying out these translations in full detail, suffice it to point out that all one has to do is replace

• the pointed quadratic space P of 8.14,9.1,9.4 by an associative composition division algebra B over F having ramification index e_{B/F} = 1,

• the pointed quadratic space Q by a composition algebra C, and the condition of Q being anisotropic by the one of C being a division algebra,

• the passage from P to ⟨⟨μ⟩⟩ ⊗ P, μ ∈ F^{×}, by the Cayley-Dickson construc-

tion Cay(B, μ). The less obvious applications of our results to composition algebras are all related, in one way or another, to the following innocuous observation, which we have not been able to extend to pointed Pfister quadratic spaces.

11.2. Proposition. Let C be a composition division algebra over F and B′ ⊆ C̄ a unital subalgebra. Assume char(F) ≠ 2 or dimF̄(B′) > 1. Then there exists a composition subalgebra B ⊆ C having ramification index e_{B/F} = 1 and satisfying B̄ = B′.

Proof. One adapts the proof of [41, Lemma 3] to the present more general setup; for completeness, we include the details. Since B′ is either a composition division algebra over F̄ or a purely inseparable field extension of characteristic 2 and exponent at most 1, it has dimension 2^{m}, m ∈ ℤ, 0 ≤ m ≤ 3. Moreover B′ is generated by m elements \overline{x_{1}}, …, \overline{x_{m}}, for some x_{1}, …, x_{m} ∈ 𝔬_{C}, where we may assume m ≥ 1 since m = 0 implies char(F) ≠ 2 by hypothesis and B := F does the job. The elements of non-zero trace in C form a Zariski open and dense subset, which therefore is open and dense in the valuation topology as well, so we may assume t_{C}(x_{1}) ≠ 0. Then B, the unital subalgebra of C generated by x_{1}, …, x_{m}, is a composition division algebra with dim_{F} (B) ≤ 2^{m}, B′ ⊆ B̄. Now Prop. 7.13 implies B̄ = B′ and e_{B/F} = 1. □

The preceding result can be refined in various ways. For example, given a composition division algebra C over F, we will exhibit (chains of) proper composition subalgebras of C having ramification index 1 and the same trace exponent as C. From this we derive normal forms for octonion and quaternion algebras over F and show that quantities subject to a few obvious constraints are the valuation data of an appropriate composition division algebra. We begin by listing a few properties of wild separable quadratic field extensions which should be well known but seem to lack a convenient reference. We therefore include the details.

11.3. Proposition. Let L be an F -algebra and suppose F̄ has characteristic 2. Then the following conditions are equivalent.

(i) L/F is a wild separable quadratic field extension and e_{L/F} = 1.

(ii) There are a positive integer r and elements α ∈ 𝔬^{×}, β ∈ 𝔬 such that β̄ ∉ F̄^{2}

and

L ≅ F [t]/(t^{2} − π^{r}αt + β).

If these conditions hold, L̄ = F̄(\sqrt{β̄}). Moreover, setting

ϑ := t mod (t^{2} − π^{r}αt + β) ∈ L (1)

in (ii), the following relations hold.

(2)  λ_{L}(γ + δϑ) = min {λ(γ), λ(δ)}  (γ, δ ∈ F),
(3)  𝔬_{L} = 𝔬1_{L} ⊕ 𝔬ϑ, 𝔭_{L} = 𝔭1_{L} ⊕ 𝔭ϑ,
(4)  texp(L) = min {e_{F} , r}.

Proof. (i) ⟹ (ii). By (i) there exists an element β ∈ 𝔬 such that β̄ ∉ F̄^{2} and L̄ = F̄(\sqrt{β̄}). Pick an element ϑ ∈ 𝔬_{L} satisfying ϑ̄ = \sqrt{β̄}. Then t_{L}(ϑ) ∈ 𝔭, and replacing β by n_{L}(ϑ) if necessary, we may assume n_{L}(ϑ) = β. Here β̄ ∉ F̄^{2} forces L = F (ϑ) = F [ϑ]. We claim there is no harm in assuming t_{L}(ϑ) ≠ 0. Indeed, for char(F) = 2, this is automatic while, if char(F) = 0, we may replace ϑ by ϑ′ := 1_{L} + ϑ. Thus, without loss, t_{L}(ϑ) ≠ 0. But this yields a unit α ∈ 𝔬^{×} such that t_{L}(ϑ) = π^{r}α, r := λ(t_{L}(ϑ)) ∈ ℤ, r > 0, and L has the form described in (ii).

(ii) ⟹ (i). By the hypotheses on β, the monic polynomial f := t^{2} − π^{r}αt + β ∈ 𝔬[t] ⊆ F [t] is irreducible over F, and L/F is a separable quadratic field extension. Define ϑ as in (1). We have n_{L}(ϑ) = β ∈ 𝔬^{×}, hence ϑ ∈ 𝔬_{L}^{×}, and ϑ̄^{2} = β̄, which implies L̄ = F̄(\sqrt{β̄}), so L is wild and has ramification index e_{L/F} = 1. Hence (i) holds.

A standard argument now yields (2), which immediately implies the remaining assertions of the proposition. □

11.4. Corollary. For a separable quadratic field extension L/F to be wild and to have ramification index 1 it is necessary and sufficient that texp(L) > 0 and there exist a trace generator u of L with \overline{n_{L}(u)} ∉ F̄^{2}. In this case, L = k[u] and u may be so chosen as to satisfy the additional relation

λ_{L}(u − u^{∗}) = texp(L). (1)

Proof. Necessity and the final statement. If L is wild and e_{L/F} = 1, we obtain texp(L) > 0 and first deal with the case e_{F} > texp(L). Then we find a positive integer r and elements α ∈ 𝔬^{×}, β ∈ 𝔬 as in Prop. 11.3 (ii) and conclude texp(L) = r from (11.3.4). Moreover, u := ϑ as defined in (11.3.1) is a trace generator of L satisfying \overline{n_{L}(u)} ∉ F̄^{2} and L = F [u]. Finally, λ_{L}(u − u^{∗}) = λ_{L}(2u − π^{r}α1_{L}), where λ_{L}(2u) = e_{F} > r = λ_{L}(π^{r}α1_{L}), which implies (1) as well. By (8.7.2), we are left with the case e_{F} = texp(L) < ∞. Then F has characteristic 0, allowing us to apply Cor. 9.14 (b) with B = P = F: there exists a scalar μ ∈ 𝔬 such that μ̄ ∉ F̄^{2} and L = F (\sqrt{μ}), so some y ∈ 𝔬_{L}^{×} has

L = F [y], t_{L}(y) = 0, \overline{n_{L}(y)} ∉ F̄^{2}. (2)

Hence u := 1_{L} + y ∈ 𝔬_{L} is a trace generator of L satisfying \overline{n_{L}(u)} ∉ F̄^{2}. Moreover, since y^{∗} = −y by (2), λ_{L}(u − u^{∗}) = λ_{L}(2y) = e_{F} = texp(L), and the proof is complete.

Sufficiency. We have L = k[u], t_{L}(u) = π^{r}α, r := texp(L), α ∈ 𝔬^{×}, and condition (ii) of Prop. 11.3 holds with β = n_{L}(u). □

11.5. Proposition. Let L be an F -algebra and suppose F̄ has characteristic 2. Then the following conditions are equivalent.

(i) L/F is a separable quadratic field extension of ramification index e_{L/F} = 2. (ii) There are a positive integer r and elements α, β ∈ 𝔬^{×} such that

L ≅ F [t]/(t^{2} − π^{r}αt + πβ).

In this case, for any prime element Π ∈ 𝔬_{L} (e.g., for

Π := t mod (t^{2} − π^{r}αt + πβ)) (1)

the following relations hold.

(2)  λ_{L}(γ + δΠ) = min \{λ(γ), λ(δ) + \frac{1}{2}\}  (γ, δ ∈ F),
(3)  𝔬_{L} = 𝔬 ⊕ 𝔬Π, 𝔭_{L} = 𝔭 ⊕ 𝔬Π,
(4)  texp(L) = min {e_{F} , r}.

Proof. Everything is standard once it has been shown in (i) ⇒ (ii) that 𝔬_{L} contains prime elements Π with t_{L}(Π) ≠ 0. But this follows from the fact that the set of elements in L with non-zero trace, by separability being open and dense in the Zariski topology, is open and dense in the valuation topology as well. □

11.6. Theorem. Let C be a composition division algebra of dimension 2^{n}, n = 2, 3, over F. Then there exists a separable quadratic subfield L ⊆ C having ramification index 1 and the same trace exponent as C: e_{L/F} = 1, texp(L) = texp(C).

Proof. Setting r := texp(C), we proceed in four steps.

1^{0}. Let us first consider the case r = 0. Then C̄ is a composition division algebra of dimension at least 2 over F̄ and hence contains a separable quadratic subfield L′ ⊆ C̄, which by Prop. 11.2 or [41, Lemma 3], lifts to a separable quadratic subfield L ⊆ C with e_{L/F} = 1, texp(L) = 0 = r. We may therefore assume from now on that r > 0, so C is wild.

2^{0}. Next we deal with the case e_{C/F} = 1. Pick a trace generator w_{0} of C, which belongs to 𝔬_{C}^{×} by (8.3.1). If w_{0} ∉ F 1_{C}, then L := F [w_{0}] is a separable quadratic subfield of C satisfying 1 ≤ e_{L/F} ≤ e_{C/F} = 1, hence e_{L/F} = 1. From 𝔭^{r} = π^{r}𝔬 = t_{L}(𝔬w_{0}) ⊆ t_{L}(𝔬_{L}) = 𝔭^{texp(L)} we conclude texp(L) ≤ r, which implies texp(L) = r by Prop. 8.2 (b). On the other hand, if w_{0} ∈ F 1_{C}, then r = e_{F} and any separable quadratic subfield L ⊆ C satisfies e_{L/F} = 1 as well as r ≤ texp(L) ≤ e_{F} = r by Prop. 8.2 (b) and (8.7.2).

3^{0}. We are left with the case e_{C/F} = 2. Then Prop. 11.2 yields a compositions subalgebra B ⊆ C with e_{B/F} = 1 and dim_{F} (B) = 2^{n−1}. If texp(B) = r, we are done for n = 2 and my apply 2^{0} to B for n = 3 to arrive at the desired conclusion. By Prop. 8.2 (b), we may therefore assume texp(B) > r. But then Thm. 9.13 justifies the assumption C = Cay(B, μ), μ = 1 − π^{d}β, for some odd integer d, 0 ≤ d < 2 texp(B), and some unit β ∈ 𝔬^{×}. In other words, we are in the situation of Prop. 9.6.

4^{0}. Applying 2^{0} to B if B is a quaternion algebra, we find a separable quadratic subfield L ⊆ B satisfying e_{L/F} = 1, texp(L) = texp(B). From Cor. 11.4 we therefore conclude that there is an element u ∈ 𝔬_{L}^{×} such that t_{L}(u) = π^{texp(B)}, \overline{n_{L}(u)} ∉ F̄^{2}. Now consider the element w := u + uΠ, which is a unit of 𝔬_{C} by (9.6.4). Moreover, setting m := \frac{d−1}{2} and observing Prop. 9.6,

t_{C}(w) = π^{r}ε, ε := 1 + π^{texp(B)−r} ∈ 𝔬^{×},
n_{C}(w) = n_{L}(u) + 2π^{−m}n_{L}(u) + πβn_{L}(u),

hence \overline{n_{C}(w)} = \overline{n_{L}(u)} ∉ F̄^{2}, so w by Prop. 11.3 generates a separable quadratic subfield L′ ⊆ C with e_{L′/F} = 1, texp(L′) = r. □

11.7. Corollary. Every composition division algebra C of dimension 2^{n}, n = 2, 3, over F contains a trace generator which is a unit in 𝔬_{C}.

Proof. Picking L ⊆ C as in Thm. 11.6, every trace generator of L is one of C and by (8.3.1) belongs to 𝔬_{L}^{×} ⊆ 𝔬_{C}^{×}. □

11.8. Corollary. Let C be a composition division algebra over F with f_{C/F} > 1. Then

texp(C) = min {λ_{C}(u − u^{∗}) | u ∈ 𝔬_{C}} = min {λ_{C}(u − u^{∗}) | u ∈ 𝔬_{C}^{×}}.

Proof. For u ∈ 𝔬_{C} we apply (2.1.6),(7.7.1),(8.7.2) and obtain

λ_{C}(u − u^{∗}) = λ_{C}(2u − t_{C}(u)1_{C}) ≥ min {e_{F} + λ_{C}(u), λ(t_{C}(u))} ≥ texp(C).

Hence it suffices to show that there is an element u ∈ 𝔬_{C}^{×} with λ_{C}(u−u^{∗}) = texp(C). If C is tame, the conjugation of C̄ cannot be the identity, so λ_{C}(u − u^{∗}) = 0 = texp(C) for some u ∈ 𝔬_{C}^{×}. We may therefore assume that C is wild. Combining the hypothesis f_{C/F} > 1 with Thm. 11.6, we find a separable quadratic subfield L ⊆ C with e_{L/F} = 1 and texp(L) = texp(C). Now Cor. 11.4 yields and element u ∈ 𝔬_{L}^{×} ⊆ 𝔬_{C}^{×} such that λ_{C}(u − u^{∗}) = λ_{L}(u − u^{∗}) = texp(L) = texp(C). □

We will see in Example 12.6 below that the hypothesis f_{C/F} > 1 in the preceding corollary cannot be avoided.

11.9. Corollary. Let C be a composition division algebra of dimension 2^{n}, n = 2, 3, over F. Then there exists a composition subalgebra B ⊆ C with

dim_{F} (B) = 2^{n−1}, e_{B/F} = 1, texp(B) = texp(C).

Proof. For n = 2, this is just Thm. 11.6, so we may assume n = 3, i.e., that C is an octonion algebra. By [41, Lemma 3], we may also assume texp(C) > 0 and, applying Thm. 11.6 again, we find a separable quadratic subfield L ⊆ C with e_{L/F} = 1, texp(L) = texp(C). Hence it suffices to prove the following lemma.

11.10. Lemma. Let C be a wild octonion division algebra over F and L ⊆ C a separable quadratic subfield such that e_{L/F} = 1, texp(L) = texp(C). Then there exists a quaternion subalgebra L ⊆ B ⊆ C with e_{B/F} = 1, texp(B) = texp(C).

Proof. Since C is wild, its residue algebra C̄ is a purely inseparable field extension of exponent 1 and degree at least 4 over F̄ containing L̄ as a quadratic subfield. Pick an element y ∈ 𝔬_{C}^{×} satisfying ȳ ∈ C̄ \ L̄. Then L and y generate a composition subalgebra B ⊆ C of dimension at most 4 whose residue algebra contains L̄(ȳ). Hence B ⊆ C is a quaternion subalgebra containing L and having e_{B/F} = 1, texp(C) = texp(L) ≥ texp(B) ≥ texp(C) by Prop. 8.2 (b). □

11.11. Normal Form Theorem: octonion algebras. For 0 < r ≤ e_{F} and an F -algebra C, the following conditions are equivalent.

(i) C is an octonion division algebra over F having trace exponent texp(C) = r. (ii) There exist

(I) a separable quadratic field extension L/F such that e_{L/F} = 1 and

texp(L) = r (in particular, L is wild), (II) a scalar α ∈ 𝔬 such that ᾱ ∉ L̄^{2},

(III) either a scalar β ∈ 𝔬 such that β̄ ∉ L̄(\sqrt{ᾱ})^{2} and
C ≅ Cay(L; α, β)

or a scalar β ∈ 𝔬^{×} such that

C ≅ Cay(L; α, 1 − πβ) or C ≅ Cay(L; α, πβ).

If this is so, then e_{C/F} = 1, C̄ ≅ L̄(\sqrt{ᾱ}, \sqrt{β̄}) in the first alternative of (III), while e_{C/F} = 2, C̄ ≅ L̄(\sqrt{ᾱ}) otherwise.

Proof. (i) ⟹ (ii). Thm. 11.6 yields a separable quadratic subfield L ⊆ C satisfying (I), and applying Lemma 11.10, we find a quaternion subalgebra L ⊆ B ⊆ C with e_{B/F} = 1, texp(B) = texp(C) = r = texp(L). Now Cor. 9.14 (b) yields a quantity α satisfying (II) and B ≅ Cay(L, α), B̄ ≅ L̄(\sqrt{ᾱ}), while (a) and (b) of the same corollary yield a quantity β satisfying (III) as well as the remaining assertions of the theorem.

(ii) ⟹ (i). This follows immediately from Prop. 9.2 and Cor. 9.14. □

Basically the same arguments, inserting the obvious simplifications at the appropriate places, leads to the following quaternionic version of the preceding result.

11.12. Normal Form Theorem: quaternion algebras. For 0 < r ≤ e_{F} and an F -algebra B, the following conditions are equivalent.

(i) B is a quaternion division algebra over F having trace exponent texp(B) = r. (ii) There exist

(I) a separable quadratic field extension L/F such that e_{L/F} = 1 and

texp(L) = r (in particular, L is wild), (II) either a scalar α ∈ 𝔬 such that ᾱ ∉ L̄^{2} and

B ≅ Cay(L, α),

or a scalar α ∈ 𝔬^{×} such that

B ≅ Cay(L; 1 − πα) or B ≅ Cay(L, πα).

If this is so, then e_{B/F} = 1, B̄ ≅ L̄(\sqrt{ᾱ}) in the first alternative of (II), while e_{B/F} = 2, B̄ ≅ L̄ otherwise. □

On the basis of the preceding results, it can now be shown that the three valuation data we are interested in can be pre-assigned in advance pretty much arbitrarily once the obvious constraints are taken into account:

11.13. Corollary. Let e, n, r be integers with n ≥ 0 and A an F̄-algebra of dimension 2^{n}. There exists a composition division algebra C over F satisfying C̄ ≅ A, e_{C/F} = e, texp(C) = r if and only if the following conditions are fulfilled.

(a) e ∈ {1, 2}, 0 ≤ n ≤ 4 − e, 0 ≤ r ≤ e_{F}, r = e_{F} (for n = 0, e = 1).

(b) A is a composition division algebra for r = 0, and A/F̄ is a purely inseparable field extension of characteristic 2 and exponent at most 1 for r > 0.

Proof. If C is a composition division algebra of dimension 2^{m}, 0 ≤ m ≤ 3, over F having the prescribed valuation data, then e = e_{C/F} ∈ {1, 2} by (7.9.8), 0 ≤ r = texp(C) ≤ e_{F} by (8.7.2) and n = m or n = m − 1 according as e_{C/F} is 1 or 2 by Prop. 7.13; moreover, n = 0 and e = 1 imply C ≅ F. Summing up and observing (8.7.1), we obtain (a), while (b) follows from Prop. 8.2 (d) combined with the remark to Prop. 10.3. Conversely, suppose (a) and (b) are fulfilled. If r = 0, the existence of a composition division algebra over F with the desired properties follows from from [41, Thms. 1,2]. We may therefore assume r > 0, which by (b) implies that A/F̄ is a purely inseparable field extension of characteristic 2, exponent at most 1 and degree 2^{n}. We first consider the case n = 0. If e = 1, then C = F has the prescribed valuation data since r = e_{F} by (a). If e = 2, Prop. 11.5 yields a separable quadratic field extension L/k having e_{L/F} = 2, texp(L) = r. We may therefore assume n > 0. By Prop. 11.3, there exists a separable quadratic field extension L/F such that e_{L/F} = 1, texp(L) = r and L̄ ⊆ A. Hence A = L̄, or A = L̄(\sqrt{ᾱ}) for some α ∈ 𝔬, ᾱ ∉ L̄^{2}, or A = L̄(\sqrt{ᾱ}, \sqrt{β̄}) for some α, β ∈ 𝔬, ᾱ ∉ L̄^{2}, β̄ ∉ L̄(\sqrt{ᾱ})^{2} according as n = 1, 2, 3. In each case, Prop. 11.3 and Thms. 11.11, 11.12 yield a composition division algebra C over F having the prescribed valuation data. □

Not surprisingly, however, the above valuation data are far away from classifying composition division algebras over 2-Henselian fields. This fact is underscored by the following example.

11.14. Example. Suppose F̄ has characteristic 2 and let F̄ ⊆ K′ ⊆ L′ be a chain of purely inseparable field extensions having exponent at most 1 over F̄ such that [L′ : F̄] = 2^{n}, [K′ : F̄] = 2^{n−1}, n = 2, 3. Furthermore, suppose r, s ∈ ℤ satisfy the relations 0 < s < r ≤ e_{F}. Then Cor. 11.13 yields a composition division algebra B over F with

e_{B/F} = 1, B̄ ≅ K′, texp(B) = r. (1)

We claim there are an infinite number of mutually non-isomorphic composition division algebras C over F containing B as a subalgebra and having e_{C/F} = 1, C̄ ≅ L′, texp(C) = s. This is in stark contrast to the fact that unramified composition division algebras up to isomorphism are uniquely determined by their residue algebras [41, Thm. 1].

To prove our claim, we fix an element β ∈ 𝔬 with L′ = K′(\sqrt{β̄}) and have β̄ ∉ B̄^{2} by (1), which implies \overline{αβ} ∉ B̄^{2} for any α ∈ 𝔬^{×} with ᾱ ∈ B̄^{2}. Setting

d := 2(r − s) = 2 (texp(B) − s) > 0, μ_{α} := 1 − π^{d}αβ ∈ 𝔬, C_{α} := Cay(B, μ_{α}),

we conclude from Prop. 9.7 that μ_{α} ∈ 𝔬^{×} and C_{α} is a composition division algebra over F satisfying

e_{C_{α}/F} = 1, \overline{C_{α}} ≅ L′, texp(C_{α}) = s.

It therefore suffices to show, for any additional element α′ ∈ 𝔬^{×} with \overline{α′} ∈ B̄^{2}, that C_{α} ≅ C_{α′} implies ᾱ = \overline{α′}. To see this, we write ᾱ = δ^{2}, \overline{α′} = δ^{′2} with δ, δ′ ∈ B̄ and invoke Cor. 9.10 (b) to derive the following chain of implications.

C_{α} ≅ C_{α′} ⟹ \overline{αβ} ≡ \overline{α′β} mod B̄^{2}
⟹ ∃ γ ∈ B̄ : δ^{2}β̄ = δ^{′2}β̄ + γ^{2}
⟹ ∃ γ ∈ B̄ : (δ − δ′)\sqrt{β̄} + γ = 0
⟹ δ = δ′ ⟹ ᾱ = \overline{α′}.
□

12. Types of composition algebras and heights.

The trace exponent has been our principal tool so far to detect wildness in pointed quadratic spaces and related objects. Other tools of this kind may be obtained by consulting the literature. It is the purpose of the present section to recast these tools in the setting of composition algebras and to compare them with the trace exponent. We begin by describing what will turn out later (see Cor. 12.4 below) to be a dichotomy of composition division algebras over a 2-Henselian field F.

12.1. Types of composition algebras. A composition division algebra C over F is said to be of unitary (resp. of primary) type if there exist an associative composition division algebra B over F with e_{B/F} = 1 and a scalar μ which is a unit (resp. a prime element) in 𝔬 such that C ≅ Cay(B, μ). We record a few easy but useful observations.

(a) By Prop. 11.2, C is of unitary or of primary type, provided f_{C/F} > 1 or char(F) ≠ 2.

(b) For C to be of primary type it is necessary by Prop. 9.2 that C have ramification index e_{C/F} = 2.

(c) If C is tame, then C is of unitary (resp. of primary) type if and only if C is unramified (resp. ramified) [41].

(d) Suppose F has characteristic 0 and F̄ has characteristic 2. A quadratic field extension of F may or may not be wild. But if it is, it is of primary (resp. of unitary) type if and only if it is tamely ramified (resp. wildly ramified or wildly unramified) in the sense of [22, p. 60].

12.2. Remark. At this stage we cannot rule out the possibility that a composition division algebra C over F is both of unitary and of primary type: conceivably, there could exist composition division algebras B, B′ over F of ramification index 1, and a unit μ as well as a prime element μ′ in 𝔬 such that Cay(B, μ) ≅ C ≅ Cay(B′, μ′). For showing that this scenario is actually impossible, the following improvement of Prop. 8.5 will play a crucial role.

12.3. Theorem. Let C be a composition division algebra over F.

(a) If C is of primary type, then ω(C) = texp(C).

(b) Consider the following conditions on C.

(i) ω(C) = texp(C) − \frac{1}{2}.

(ii) There are trace generators of C belonging to 𝔭_{C}. (iii) C is wild of unitary type and ramification index 2.

Then the implications

(i) ⟺ (ii) ⟸ (iii)

hold. Moreover, if f_{C/F} > 1 or char(F) ≠ 2, then all three conditions are equivalent.

Proof. We begin with the first part of (b).

(i) ⟹ (ii). Let w_{0} be a regular trace generator of C. Then (i) and (8.5.1) show λ_{C}(w_{0}) > 0, hence w_{0} ∈ 𝔭_{C}.

(ii) ⟹ (i). Let w_{0} ∈ 𝔭_{C} be a trace generator of C. Then ω(C) ≤ λ(t_{C}(w_{0})) − λ_{C}(w_{0}) < texp(C), and Prop. 8.5 (a) gives (ii).

(iii) ⟹ (ii). If C is wild of unitary type and ramification index 2, we have C = Cay(B, μ), B an associative composition division algebra over F with e_{B/F} = 1, μ ∈ 𝔬^{×}, and Thm. 9.13 shows that we are in the situation of Prop. 9.6 with Q = C, P = B (cf. the translation formalism 11.1). Picking a trace generator u of B, we apply (9.6.1),(9.6.5) to obtain

λ(t_{C}(uΠ)) = λ(t_{C}(π^{− \frac{d−1}{2}} (u + uj))) = λ(t_{B}(u)) − \frac{d − 1}{2}
= texp(B) − \frac{d − 1}{2} = texp(C),

so uΠ ∈ 𝔭_{C} is a trace generator of C, showing (ii). Before completing the proof of (b), we turn to

(a) By hypothesis, we are in the situation of Prop. 9.2 with Q = C, P = B as before, and by (9.2.2), an element of 𝔭_{C} has the form y = u + vj, u ∈ 𝔭_{B}, v ∈ 𝔬_{B}. Hence π^{−1}u ∈ 𝔬_{B} since e_{B/F} = 1, and from (9.2.3) we conclude

λ(t_{C}(y)) = λ(t_{B}(u)) = λ(t_{B}(π^{−1}u)) + 1 ≥ texp(B) + 1 = texp(C) + 1 texp(C).

Thus 𝔭_{C} does not contain trace generators of C, violating (ii), hence (i), in (b). Now Prop. 8.5 (a) implies (a).

It remains to show (i) ⇒ (iii) in (b) under the assumption f_{C/F} > 1 or char(F) ≠ 2. Then e_{C/F} = 2 by Prop. 8.5 (c), and texp(C) = ω(C) + \frac{1}{2} > 0 by (i) forces C to be wild. Moreover, by 12.1 (a), it is of unitary or of primary type, the latter case being excluded by (i) and (a). □

12.4. Corollary. A composition division algebra C over F cannot be both of unitary and of primary type.

Proof. If C is tame, the assertion follows immediately from 12.1 (c). If C is wild of ramification index 1, it cannot be of primary type by 12.1 (b). Hence we are left with the case that C is wild of ramification index 2. Then, by Thm. 12.3, ω(C) = texp(C) is an integer if C is of primary type, while ω(C) = texp(C) − \frac{1}{2} is not if C is of unitary type. □

12.5. Corollary. Let C be a composition division algebra of primary type over F and suppose B ⊆ C is a composition subalgebra with

e_{B/F} = 1, dim_{k}(B) = \frac{1}{2}dim_{k}(C).

Then texp(B) = texp(C).

Proof. Since C is not of unitary type by Cor. 12.4, we obtain C ≅ Cay(B, μ) for some prime element μ ∈ 𝔬. Hence texp(B) = texp(C) by (9.2.3). □

Both Cor. 11.8 and Thm. 12.3 (b) fail in the exceptional cases stated therein. This is the upshot of the following example.

12.6. Example. Let L/F be a separable quadratic field extension of ramification index 2 and suppose char(F̄) = 2. Then we are in the situation of of Prop. 11.5, and if r ≤ e_{F}, then texp(L) = r by (11.5.4). Moreover, (11.5.1) gives λ(t_{L}(Π)) = λ(π^{r}α) = r = texp(L), so Π ∈ 𝔭_{L} is a trace generator of L; in particular, for char(F) = 2, parts (i),(ii) of Thm. 12.3 (b) hold but (iii) doesn’t. On the other hand, if r > e_{F}, then F has characteristic zero and

L = F (\sqrt{πγ}) = Cay(F, πγ), γ := −β + \frac{π^{2r−1}}{4} α^{2} ∈ 𝔬^{×},

is of primary type, forcing ω(L) = texp(L) by Thm. 12.3 (a).

Let u = γ + δΠ ∈ 𝔬_{L}, γ, δ ∈ 𝔬 (cf. (11.5.3)). Applying (11.5.2),

λ_{L}(u − u^{∗}) = λ_{L}(−π^{r}αδ + 2δΠ) = min \{r + λ(δ), e_{F} + λ(δ) + \frac{1}{2}\}
≥ min \{r, e_{F} + \frac{1}{2}\},

and this minimum is attained for, e.g., u = Π. Thus, by(11.5.4),

min \{λ_{L}(u − u^{∗}) | u ∈ 𝔬_{L}\} = \begin{cases}texp(L) & for r ≤ e_{F}, \\ texp(L) + \frac{1}{2} & for r > e_{F} , \end{cases}

so in the latter case, the conclusion of Cor. 11.8 does not hold.

We remark in closing that Cor. 11.8 also fails if L/F is tame of ramification index 2 since this implies texp(L) = 0 while ^{∗} induces the identity on L̄, so λ_{L}(u−u^{∗}) > 0 for all u ∈ 𝔬_{L}.

12.7. Remark. There is an alternate way of proving Cor. 12.4, by working with quadratic forms. Let C be a composition division algebra over F and suppose C is of unitary and of primary type. Then there are composition division algebras B, B′ over F of ramification index 1 and a unit μ as well as a prime element μ′ in 𝔬 such that

Cay(B, μ) ≅ C ≅ Cay(B′, μ′).

Then Prop. 3.12 yields a scalar γ ∈ F^{×} satisfying

(1) Cay(B, μ) ≅ Cay(B, γ), Cay(B′, μ′) ≅ Cay(B′, γ)

Since e_{B/F} = e_{B′/F} = 1, we conclude from (9.1.2) that λ(n_{B}(B^{×})) = λ(n_{B′}(B^{′×})) = 2ℤ. Hence Prop. 7.5 (b) and the first relation of (1) show that λ(γ) is even, while Prop. 7.5 (b) and the second relation of (1) show that λ(γ) is odd, a contradiction.

With the aim of generalizing Thm. 11.6, Cor. 11.9 and Lemma 11.10, we next turn to the problem of finding (chains of) subalgebras having ramification index 1 and pre-assigned trace exponents. Once the obvious constraints are taken into account (provided, e.g., by Prop. 8.2 (b) and Cor. 12.5), we will show that chains of such subalgebras always exist.

12.8. Theorem. Suppose F̄ has characteristic 2 and let C be a composition division algebra of dimension 2^{n}, n = 2, 3, over F that is not a quaternion division algebra of primary type. Given r ∈ ℤ, texp(C) ≤ r ≤ e_{F}, there exists a separable quadratic subfield L ⊆ C with e_{L/F} = 1, texp(L) = r.

Proof. The case r = texp(C) having been settled by Thm. 11.6, we may assume r > texp(C). Applying Cor. 11.9 and then Thm. 11.6, we find a composition subalgebra B ⊆ C and a separable quadratic subfield K ⊆ B with

dim_{F} (B) = 2^{n−1}, e_{K/F} = e_{B/F} = 1, texp(K) = texp(B) = texp(C).

Suppose for the time being that the case n = 2 has been solved and let n = 3. The quaternion algebra B, having ramification index 1, cannot be of primary type, allowing us to apply the case n = 2 to B in place of C and leading us to the desired conclusion.

We are thus reduced to the case n = 2, which we will assume for the rest of the proof. Then K = B and C has dimension 4, hence is not of primary type by hypothesis. We are therefore lead to a unit μ ∈ 𝔬^{×} with

C = Cay(K, μ) = K ⊕ Kj. (1)

Let us first assume texp(K) = texp(C) = 0, so K is tame. Since C is a division algebra, we conclude μ̄ ∉ nK̄(K̄^{×}) from Prop. 9.3. In particular, we have μ̄ ∉ F̄^{2}. The hypothesis texp(K) = 0 yields an element v ∈ 𝔬_{K} having t_{K}(v) = 1. Observing (1), we put

w := π^{r}v + j ∈ C

and obtain t_{C}(w) = π^{r}, n_{C}(w) = π^{2r}n_{K}(v) − μ, \overline{n_{C}(w)} = μ̄ ∉ F̄^{2}. In particular, w ∈ 𝔬_{C}^{×} \ F 1_{C} and we conclude from Prop. 11.3 that L = F [w] ⊆ C is a separable quadratic subfield of the desired kind.

We are left with the case texp(K) = texp(C) > 0. Then K is wild, and Cor. 11.4 yields an element u ∈ 𝔬_{K}^{×} with

t_{K}(u) = π^{texp(K)}, \overline{n_{K}(u)} ∉ F̄^{2}. (2)

By (1) and Cor. 9.14, we may assume

(3)  μ̄ ∉ F̄^{2} or μ = 1 − πβ for some β ∈ 𝔬^{×}

according as C has ramification index 1 or 2. We now put

w := \begin{cases}π^{r−texp(K)}u + j ∈ C & if e_{C/F} = 1, \\ π^{r−texp(K)}u + uj ∈ C & if e_{C/F} = 2. \end{cases}

Then t_{C}(w) = π^{r} by (2), and for e_{C/F} = 1 we obtain \overline{n_{C}(w)} = μ̄ ∉ F̄^{2} by (3). Similarly, for e_{C/F} = 2, we obtain \overline{n_{C}(w)} = \overline{n_{K}(u)} ∉ F̄^{2} by (2),(3). In either case, w ∈ 𝔬_{C}^{×} \ k1_{C} generates a separable quadratic subfield L := F [w] ⊆ C of ramification index e_{L/F} = 1 and trace exponent texp(L) = r (Prop. 11.3). □

12.9. Corollary. Suppose F̄ has characteristic 2 and let C be an octonion division algebra over F that is not of primary type. Given r, s ∈ ℤ, texp(C) ≤ r ≤ s ≤ e_{F}, there exists a filtration L ⊆ B ⊆ C consisting of a quaternion subalgebra B ⊆ C and a separable quadratic subfield L ⊆ B such that e_{L/F} = e_{B/F} = 1 and texp(B) = r, texp(L) = s.

Proof. It suffices to construct a quaternion subalgebra B ⊆ C with e_{B/F} = 1, texp(B) = r because Thm. 12.8 applies to such a B and also yields an L with the desired properties.

To construct B, we first invoke Thm. 11.6 and Cor. 11.9 to find a quaternion subalgebra B_{1} ⊆ C and a separable quadratic subfield L_{1} ⊆ B_{1} satisfying

(1) e_{L_{1}/F} = e_{B_{1}/F} = 1, texp(L_{1}) = texp(B_{1}) = texp(C).

By hypothesis, (1) and Cor. 9.14 yield units μ_{1}, μ_{2} ∈ 𝔬^{×} with

(2)  B_{1} = Cay(L_{1}, μ_{1}) = L_{1} ⊕ L_{1}j_{1}, \overline{μ_{1}} ∉ F̄^{2},
C = Cay(B_{1}, μ_{2}) = B_{1} ⊕ B_{1}j_{2},

where

\overline{μ_{2}} ∉ F̄^{2} for e_{C/F} = 1 and μ_{2} = 1 − πβ_{2}, β_{2} ∈ 𝔬^{×} for e_{C/F} = 2.

By the same token,

B_{2} := Cay(L_{1}, μ_{2}) = L_{1} ⊕ L_{1}j_{2} ⊆ C

is a quaternion subalgebra not of primary type by Cor. 12.4 with texp(B_{2}) = texp(L_{1}) = texp(C) ≤ r ≤ e_{F}, e_{B_{2}/F} = e_{C/F}. Hence Thm. 12.8 leads us to a separable quadratic subfield L ⊆ B_{2} with e_{L/F} = 1, texp(L) = r and (2) combines with Cor. 9.14 (b) to show that the quaternion subalgebra

B = Cay(L, μ_{1}) = L ⊕ Lj_{1} ⊆ C

satisfies e_{B/F} = 1, texp(B) = texp(L) = r. □

12.10. Heights. Over a Henselian field having residual characteristic p > 0, the height as an important invariant of a central associative division algebra of degree p over F has been considered by Saltman [45, pp. 1757, 1765-6] (who uses the term “level”), Kato [23, § 1] (who calls it the “ramification number”), and Tignol [51, 3.2]. It is, in particular, Tignol’s approach that suggests two immediate translations to the setting of composition algebras.

Let C be a composition division algebra over our 2-Henselian field F. We use the maps h_{com} : C^{×} × C^{×} → ℚ_{∞}, h_{ass} : C^{×} × C^{×} × C^{×} → ℚ_{∞} given by

(1)  h_{com}(x, y) := λ_{C}([x, y]) − λ_{C}(x) − λ_{C}(y) ≥ 0,
(2)  h_{ass}(x, y, z) := λ_{C}([x, y, z]) − λ_{C}(x) − λ_{C}(y) − λ_{C}(z) ≥ 0

for all x, y, z ∈ C^{×} to define

(3)  hgt_{com}(C) := inf {h_{com}(x, y) | x, y ∈ C^{×}},
(4)  hgt_{ass}(C) := inf {h_{ass}(x, y, z) | x, y, z ∈ C^{×}}

and to call these numbers the commutative height and the associative height of C, respectively. If C has dimension at most 2, then hgt_{com}(C) = hgt_{ass}(C) = ∞. On the other hand, if C is a quaternion algebra, then hgt_{com}(C) < ∞ = hgt_{ass}(C) and hgt_{com}(C) agrees with what Tignol calls its height, while if C is an octonion algebra, its commutative and its associative height are both finite.

A general theorem of Tignol [51, 3.12] implies hgt_{com}(C) = ω(C) for any quaternion division algebra C over F. This special observation is part of a much more general picture that will be summarized in the following theorem, whose proof in the quaternionic case is independent of [51] and, in fact, works uniformly in the octonionic case as well.

12.11. Theorem. If C is a quaternion division algebra over F, then

hgt_{com}(C) = ω(C).

If C is an octonion division algebra over F, then

hgt_{com}(C) = hgt_{ass}(C) = ω(C).

Proof. Let C be a composition division algebra of dimension 2^{n}, n = 2, 3, over F. We must show

(1)  hgt_{com}(C) = ω(C),
(2)  hgt_{com}(C) = hgt_{ass}(C) = ω(C)  (for n = 3).

To do so, we combine Thm. 2.8 with (10.10.1) to obtain

λ_{C}([x_{1}, x_{2}, x_{3}]) ≥ ω(C) + λ_{C}(x_{1}) + λ_{C}(x_{2}) + λ_{C}(x_{3})

for all x_{1}, x_{2}, x_{3} ∈ C. Combining this and (10.10.2) with (12.10.1−4), we conclude

(3) hgt_{com}(C) ≥ ω(C), hgt_{ass}(C) ≥ ω(C).

To complete the proof of (1),(2), it therefore suffices to show that

(4)  h_{com}(x_{1}, x_{2}) = ω(C)  (for some x_{1}, x_{2} ∈ C),
(5)  h_{ass}(x_{1}, x_{2}, x_{3}) = ω(C)  (for n = 3 and some x_{1}, x_{2}, x_{3} ∈ C).

For this purpose, we require two additional formulas: suppose C = Cay(B, μ) = B⊕Bj is a Cayley-Dickson construction as in 1.10, for some associative composition algebra B over F and some scalar μ ∈ F^{×}. Then a straightforward application of (1.10.1) yields

(6)  [u, j] = (u − u^{∗})j  (u ∈ B),
(7)  [u_{1}, u_{2}, j] = [u_{1}, u_{2}]j  (u_{1}, u_{2} ∈ B).

In order to establish (4),(5), we distinguish the following cases.

Case 1. e_{C/F} = 1. Then ω(C) = texp(C) by Prop. 8.5 (c). If C is tame, the C̄ is a composition division algebra of dimension 2^{n} over F̄, so there are x_{1}, x_{2}, x_{3} ∈ 𝔬_{C}^{×} with [x_{1}, x_{2}] ∈ 𝔬_{C}^{×}, and even [x_{1}, x_{2}, x_{3}] ∈ 𝔬_{C}^{×} for n = 3. By (12.10.1),(12.10.2), this implies h_{com}(x_{1}, x_{2}) = 0 = texp(C), and even h_{ass}(x_{1}, x_{2}, x_{3}) = 0 = texp(C) for n = 3, proving (4),(5) in the tame case.

If C is wild, it must be of unitary type since e_{C/F} = 1, and Thm. 9.13 combined with Prop. 11.2 implies C = Cay(B, μ), B an associative composition division algebra over F with e_{B/F} = 1, μ = 1 − π^{d}β, d ∈ ℤ even, 0 ≤ d < 2 texp(B), β ∈ 𝔬, β̄ ∉ B̄^{2}. In particular, taking into account 11.1, we are in the situation of Prop. 9.7. Applying Cor. 11.8, we find an element u ∈ 𝔬_{B}^{×} such that λ_{B}(u − u^{∗}) = texp(B). Hence (9.7.2),(9.7.3),(9.7.6) and (6) yield

h_{com}(u, Ξ) = λ_{C}([u, Ξ]) − λ_{C}(u) − λ_{C}(Ξ) = λ_{C}(π^{− \frac{d}{2}} [u, j])
= λ_{B}(u − u^{∗}) − \frac{d}{2} = texp(B) − \frac{d}{2} = texp(C).

Thus (4) holds for x_{1} = u, x_{2} = Ξ, and we have established (1) in Case 1.

If n = 3 (C still assumed to be wild), Case 1 applies to B, so (4) yields elements u_{1}, u_{2} ∈ B such that h_{com}(u_{1}, u_{2}) = texp(B). Hence (7) and (9.7.6) imply

h_{ass}([u_{1}, u_{2}, Ξ]) = λ_{C}(π^{− \frac{d}{2}} [u_{1}, u_{2}, j]) − λ_{B}(u_{1}) − λ_{B}(u_{2})
= h_{com}(u_{1}, u_{2}) − \frac{d}{2} = texp(B) − \frac{d}{2} = texp(C),

giving (5) for x_{1} = u_{1}, x_{2} = u_{2}, x_{3} = Ξ, and settling Case 1 completely.

Case 2. e_{C/F} = 2. If C is of primary type, then ω(C) = texp(C) by Thm. 12.3 (a), and C = Cay(B, μ) with B as before and μ a prime element in 𝔬. Applying Thm. 9.13 (a) and picking u ∈ 𝔬_{B}^{×} with texp(C) = texp(B) = λ_{B}(u − u^{∗}), we obtain

h_{com}(u, j) = λ_{C}([u, j]) − λ_{B}(u) − λ_{C}(j)
= λ_{C}((u − u^{∗})j) − λ_{C}(j) = λ_{B}(u − u^{∗}) = texp(C),

and (4) holds for x_{1} = u, x_{2} = j. Moreover, for n = 3, Case 1 applies to B and yields u_{1}, u_{2} ∈ B^{×} having h_{com}(u_{1}, u_{2}) = texp(B) = texp(C), allowing us to compute

h_{ass}(u_{1}, u_{2}, j) = λ_{C}([u_{1}, u_{2}]j) − λ_{B}(u_{1}) − λ_{B}(u_{2}) − λ_{C}(j)
= h_{com}(u_{1}, u_{2}) = texp(C)

and completing the proof of (5) for x_{1} = u_{1}, x_{2} = u_{2}, x_{3} = j.

Now suppose C is of unitary type. Then C is wild since e_{C/F} = 2, and Thm. 12.3 (b) shows ω(C) = texp(C) − \frac{1}{2}. This time, Thm. 9.13 implies C = Cay(B, μ) with B as before, μ = 1 − π^{d}β, d ∈ ℤ odd, 0 ≤ d < 2 texp(B), β ∈ 𝔬^{×}, so in view of 11.1 we are in the situation of Prop. 9.6. Picking again an element u ∈ 𝔬_{B}^{×} with texp(B) = λ_{B}(u − u^{∗}), we obtain

h_{com}(u, Π) = λ_{C}([u, Π]) − λ_{B}(u) − λ_{C}(Π) = λ_{C}(π^{− \frac{d−1}{2}} [u, j]) − \frac{1}{2}
= λ_{C}((u − u^{∗})j) − \frac{d − 1}{2} − \frac{1}{2} = λ_{B}(u − u^{∗}) − \frac{d − 1}{2} − \frac{1}{2}
= texp(B) − \frac{d − 1}{2} − \frac{1}{2} = texp(C) − \frac{1}{2} = ω(C),

and (4) holds for x_{1} = u, x_{2} = Π. If, in addition, n = 3, then Case 1 applies to B, yielding u_{1}, u_{2} ∈ B^{×} with h_{com}(u_{1}, u_{2}) = texp(B). Hence

h_{ass}(u_{1}, u_{2}, Π) = λ_{C}([u_{1}, u_{2}, Π]) − λ_{B}(u_{1}) − λ_{B}(u_{2}) − λ_{C}(Π)
= λ_{C}(π^{− \frac{d−1}{2}} [u_{1}, u_{2}, j]) − λ_{B}(u_{1}) − λ_{B}(u_{2}) − \frac{1}{2}
= λ_{C}([u_{1}, u_{2}]j) − \frac{d − 1}{2} − λ_{B}(u_{1}) − λ_{B}(u_{2}) − \frac{1}{2}
= h_{com}(u_{1}, u_{2}) − \frac{d − 1}{2} − \frac{1}{2} = texp(B) − \frac{d − 1}{2} − \frac{1}{2}
= texp(C) − \frac{1}{2} = ω(C).  □

Part III. Connections with K-theory

13. Introduction to Part III

13.1. The goal of this part of the paper is to translate the results of Part II into the language of Kato’s filtration on Milnor K-theory mod 2. Because it costs little extra, we will also give a dictionary relating traditional valuation-theoretic terms on associative division algebras of prime degree p with Milnor K-theory mod p.

For the remainder of the paper, we fix a field F of characteristic zero that has a Henselian discrete valuation λ and residue field \overline{F} of prime characteristic p. We assume that F contains a primitive p-th root of unity ζ and set

m := p · \frac{λ(p)}{p − 1}.

This is an integer divisible by p because λ(p)/(p − 1) = λ(ζ − 1), see, e.g., [10, 4.1.2(i)].

Recall that the Milnor K-ring of F, denoted K_{∗}^{M}(F), is the tensor algebra (over ℤ) of the abelian group F^{×} modulo the “Steinberg relation” a ⊗ (1 − a) = 0 for a ∈ F, a ≠ 0, 1.

One writes {a_{1}, …, a_{q}} for the image of a_{1} ⊗ ⋯ ⊗ a_{q} in K_{q}^{M}(F). We put k_{q}(F) for K_{q}^{M}(F)/p, and we abuse notation by writing {a_{1}, …, a_{q}} also for the image of that element in k_{q}(F); such a class in k_{q}(F) is called a symbol. See, e.g., [19] for basic properties. Kato, Bloch, and Gabber proved that k_{q}(F) is isomorphic to H^{q}(F, μ_{p}^{⊗q}) via the “Galois symbol”, which sends {a_{1}, …, a_{q}} ↦ (a_{1}) · (a_{2}) ⋯ (a_{q}); this identifies nonzero symbols in k_{q}(F) with nonzero symbols in H^{q}(F, μ_{p}^{⊗q}). We are mainly interested in the following cases:

(1) \underline{q = 1}: k_{1}(F) and H^{1}(F, μ_{p}) are naturally identified with F^{×}/F^{×p}. A nonzero element xF^{×p} defines a degree p extension F (χ) such that χ^{p} = x.

(2) \underline{q = 2}: H^{2}(F, μ_{p}^{⊗2}) is identified (via ζ) with H^{2}(F, μ_{p}), i.e., the p-torsion in the Brauer group of F. We fix the identification with the Brauer group so that the nonzero symbol {x, y} in k_{2}(F) is sent to the associative central division F -algebra of dimension p^{2} generated by elements χ, ψ satisfying

χ^{p} = x, ψ^{p} = y, and χψ = ζψχ.

(3) \underline{p = 2}: In this case, there is a bijection between symbols in k_{q}(F) and q-Pfister (quadratic) forms given by sending {a_{1}, …, a_{q}} to ⟨⟨a_{1}, …, a_{q}⟩⟩. For q ≤ 3, we can of course further identify anisotropic q-Pfister forms with composition algebras of dimension 2^{q}.

For a nonzero symbol γ ∈ k_{q}(F) from cases (1) or (2), write D for the corresponding division F -algebra. As the valuation λ is Henselian, it extends to a discrete valuation λ_{D} on D via the usual formula λ_{D}(x) := λ(N_{F (x)/F} (x))/[F (x) : F], cf. (7.14.1). The definition of residue division algebra \overline{D}, ramification index e_{D/F}, etc., is the same as for quaternion algebras, and the fundamental relation of Prop. 7.13 holds, see [54, p. 393] for references.

13.2. Below, we will recall the filtration on k_{q}(F) and define invariants e_{γ} (= 1 or p) and depth(γ) for a symbol γ ∈ k_{q}(F) in terms of K-theory. We will prove K-theoretic analogues of the Local Norm Theorem 8.10 (§15), the Normal Form Theorems 11.11 and 11.12 (§16), and Theorem 9.9 (§18). These proofs use the background results on quadratic forms over Henselian fields from §7 but not the deeper results from the rest of Part II.

In the final sections of the paper (§§19–22) we give a dictionary between properties of symbols in k_{q}(F) in the cases q = 1, q = 2, or p = 2. The proofs in the case p = 2 rely heavily on the full strength of the results in Part II.

14. The filtration on K-theory

We now recall the Kato filtration on Milnor K-theory over F, together with the isomorphisms of the graded components with various modules of differential forms, etc., over \overline{F}.

14.1. Filtration and depth. Write 𝔬 for the valuation ring on F and 𝔭 for its maximal ideal. One can filter 𝔬 as

𝔬 \ {0} = U_{0} ⊇ U_{1} ⊇ U_{2} ⊇ ⋯

where U_{i} := 1 + 𝔭^{i} for i ≥ 1.

For q ≥ 1, setting U^{i}k_{q}(F) to be additively generated by elements {u} · k_{q−1}(F) for u ∈ U_{i} defines a filtration

k_{q}(F) = U^{0}k_{q}(F) ⊇ U^{1}k_{q}(F) ⊇ ⋯

For i > m, U_{i} consists of p-th powers [10, 4.1.2(ii)], so U^{i}k_{q}(F) is zero.

The depth of γ ∈ k_{q}(F) is the supremum of {i | γ ∈ U^{i}k_{q}(F)}. The only element of depth > m is zero, which has depth ∞. The filtration is compatible with the product in the sense that U^{i}k_{r}(F) · U^{j}k_{s}(F) ⊆ U^{i+j}k_{r+s}(F) by [10, 4.1.1b]. Said differently, for elements α ∈ k_{r}(F) and β ∈ k_{s}(F), we have

(1)  depth(α) + depth(β) ≤ depth(α · β);

this inequality can be strict, see Example 20.4.

14.2. Kato isomorphisms. For nonzero γ ∈ k_{q}(F) with q ≥ 1 of depth d, we consider the (nonzero) image of γ in gr^{d} k_{q}(F) := U^{d}k_{q}(F)/U^{d+1}k_{q}(F); this is the initial form of γ. The results of Kato, et al, include specific isomorphisms:

gr^{d} k_{q}(F) ≅ \begin{cases}k_{q}(\overline{F}) ⊕ k_{q−1}(\overline{F}) & if d = 0; \\ Ω^{q−1} & if 0 < d < m and p does not divide d; \\ \frac{Ω^{q−1}}{Z^{q−1}} ⊕ \frac{Ω^{q−2}}{Z^{q−2}} & if 0 < d < m and p divides d; \\ H^{1}(\overline{F} , ν(q − 1)) ⊕ H^{1}(\overline{F} , ν(q − 2)) & if d = m. \end{cases}

Here Ω^{1} denotes the \overline{F} -vector space of derivations \overline{F} → \overline{F}, Ω^{q} := ∧^{q}Ω^{1} for q ≥ 1, Ω^{0} = \overline{F} and Ω^{−1} = \{0\}. The subspace Z^{q} is the kernel of the differential Ω^{q} → Ω^{q+1}, i.e., Z^{q} is the subspace of exact forms. The groups ν(q) are defined in terms of the Cartier operator [10, pp. 4,5]; they are chosen so that H^{1}(\overline{F} , ν(q − 1)) in characteristic p plays the role of the Galois cohomology group H^{q}(K, μ_{p}^{⊗(q−1)}) for K of characteristic ≠ p.

We refer to these isomorphisms as the “Kato isomorphisms”. Fix a uniformizer π for λ and write a_{i} and b for elements of 𝔬^{×}. The isomorphisms are:

d  map
d = 0  {a_{1}, a_{2}, …, a_{q}}↦({ā_{1}, …, ā_{q}}, 0)
{π, a_{1}, …, a_{q−1}}↦(0, {ā_{1}, …, ā_{q−1}})

p does not divide d {1 + bπ^{d}, a_{1}, …, a_{q−1}}↦ b̄\frac{dā_{1}}{ā_{1}} ∧ ⋯ ∧ \frac{dā_{q−1}}{ā_{q−1}}

p divides d  \{1 + bπ^{d}, a_{1}, …, a_{q−1}\}↦(b̄\frac{dā_{1}}{ā_{1}} ∧ ⋯ ∧ \frac{dā_{q−1}}{ā_{q−1}} , 0)
and d ≠ 0, m  \{π, 1 + bπ^{d}, a_{1}, …, a_{q−2}\}↦(0, b̄\frac{dā_{1}}{ā_{1}} ∧ ⋯ ∧ \frac{dā_{q−2}}{ā_{q−2}} )
d = m  \{1 + b(ζ − 1)^{p}, a_{1}, …, a_{q−1}\}↦(b̄\frac{dā_{1}}{ā_{1}} ∧ ⋯ ∧ \frac{dā_{q−1}}{ā_{q−1}} , 0)
\{π, 1 + b(ζ − 1)^{p}, a_{1}, …, a_{q−2}\}↦(0, b̄\frac{dā_{1}}{ā_{1}} ∧ ⋯ ∧ \frac{dā_{q−2}}{ā_{q−2}} )

The description of gr^{0} k_{q}(F) is a result of Bass-Tate that holds without restriction on the characteristic of \overline{F}. For depth m, of course (ζ − 1)^{p} has value m, and in case p = 2 the expression (ζ − 1)^{p} is 4.

14.3. We mention for later reference a useful fact about k_{q}(\overline{F}) for q ≥ 1. As with any field, there is a group homomorphism K_{q}^{M}(\overline{F}) → Ω^{q} defined by \{x_{1}, …, x_{q}\} ↦ \frac{dx_{1}}{x_{1}} ∧ ⋯ ∧ \frac{dx_{q}}{x_{q}} (one checks the Steinberg relation). But \overline{F} has characteristic p, so this homomorphism induces a homomorphism ψ: k_{q}(\overline{F}) → Ω^{q}. Moreover, ψ is injective by [4, 2.1] or [19, 9.7.1]. In summary, we have: for x_{1}, …, x_{q} ∈ \overline{F}^{×}, the following are equivalent:

(i) The symbol \{x_{1}, …, x_{q}\} is zero in k_{q}(\overline{F}).

(ii) dx_{1} ∧ ⋯ ∧ dx_{q} is zero in Ω^{q}.
(iii) The elements \sqrt[p]{x_{1}}, …, \sqrt[p]{x_{q}} are not p-free over \overline{F}.

The equivalence of (ii) and (iii) is [6, §V.13.2, Th. 1]. For further statements along these lines, see e.g. [20, 8.1].

Given y, x_{1}, …, x_{q} ∈ \overline{F}^{×} with dx_{1}∧⋯∧dx_{q} ≠ 0 in Ω^{q}, the preceding equivalence implies that y dx_{1} ∧ ⋯ ∧ dx_{q} = 0 in Ω^{q}/Z^{q} if and only if y is a p-th power in \overline{F} (\sqrt[p]{x_{1}}, …, \sqrt[p]{x_{q}}).

15. The Local Norm Theorem 8.10 revisited

We now prove an analogue of the Local Norm Theorem 8.10. We continue the notation of 13.1, and focus on a nonzero symbol γ ∈ k_{q}(F) where q = 1, q = 2, or p = 2. The symbol γ corresponds to a Galois field extension of F of degree p, a (central) associative division algebra of dimension p^{2} over F, or a q-Pfister quadratic form over F. Write V for the underlying vector space, which has dimension p^{q}. In each case, there is a canonical choice of homogeneous polynomial f : V → F of degree p and representing 1: the norm, the reduced norm, or the quadratic form itself. Further, the valuation λ extends to a valuation λ_{V} on V via the formula λ_{V} (v) := λ(f(v))/p, and we write 𝔬_{V}^{×} for the set of v ∈ V with value zero.

For a given a ∈ 𝔬^{×}, we ask: How close is f to representing a? Imitating the definition in 8.8, we put

nexp_{γ}(a) := sup {λ(a − f(v)) | v ∈ 𝔬_{V}^{×}} .

It is obviously equivalent to define nexp_{γ}(a) to be the supremum of all d ≥ 0 such that there exist β ∈ 𝔬 and v ∈ 𝔬_{V}^{×} such that a = (1 − π^{d}β) f(v).

15.1. Example. As f represents 1,

depth{a} ≤ nexp_{γ}(a) (a ∈ 𝔬^{×}).

Moreover, we have equality in case p = 2 and q = 0 (so necessarily f is the quadratic form ⟨1⟩ and γ = 1 ∈ ℤ/2ℤ = k_{0}(F)).

15.2. Local Norm Theorem. Suppose p = 2 or 1 ≤ q ≤ 2. For a ∈ 𝔬^{×} and a symbol γ ∈ k_{q}(F), we have:

(1)  nexp_{γ}(a) + depth γ ≤ depth({a} · γ).

Furthermore, the following are equivalent:

(i) {a} · γ = 0. (ii) a ∈ f(V). (iii) nexp_{γ}(a) > m − depth γ. (iv) nexp_{γ}(a) = ∞.

The number m was defined in 13.1 to be p λ(p)/(p − 1). Note that the inequality (15.2.1) apparently strengthens (14.1.1) by Example 15.1.

Proof. (i) and (ii) are known to be equivalent: for q = 1, it is [19, 4.7.5] and for p = 2 it is Prop. 7.5. For q = 2, the implication (ii) ⇒ (i) is elementary and the converse is due to Merkurjev-Suslin [34, 12.2].

Assume (ii), i.e., f(v) = a for some v ∈ V. Then λ_{V} (v) = λ(a)/p = 0, so v ∈ 𝔬_{V}^{×} and (iv) is obvious, hence also (iii).

We now prove equation (1). Suppose a = (1 − π^{d}β) f(v) for some d ≥ 0, β ∈ 𝔬 and v ∈ 𝔬_{V}^{×}. Then

{a} · γ = {f(v)} · γ + {1 − π^{d}β} · γ.

But the first term on the right side is zero by the equivalence of (i),(ii) already established. Hence

depth({a} · γ) = depth({1 − π^{d}β} · γ) ≥ d + depth γ,

and we have proved (1).

Finally, suppose (iii). By (1), the symbol {a} · γ has depth > m, hence the symbol is zero, proving (i). □

In order to compare this result with the Local Norm Theorem 8.10, we must relate ω with depth; for the purposes of this discussion, let us focus on the case p = 2 and put ω(γ) := m − (depth γ)/2. (This agrees with the definition of ω for composition algebras in case additionally q = 2 or 3 by Cor. 19.3(ii) below.) Translating Equation 15.2.1 into this notation gives:

nexp_{γ}(a) ≤ 2 ω(γ) − 2 ω({a} · γ).

That is, Theorem 15.2 sharpens Theorem 8.10.

15.3. Remark. If every finite extension of F has dimension a power of p (“F is p-special”) for some prime p, then Theorem 15.2 holds for that prime p and all q ≥ 1 if one adjusts slightly the statement of (ii). The adjusted statement should be in terms of a norm variety for γ as is obvious from [48, Prop. 2.4]; we leave the details to the reader. The paper [48] also provides the proof of the equivalence of (i) and the adjusted form of (ii).

16. Gathering the depth

We maintain the notation of 13.1. Recall that U^{d}k_{q}(F) is generated as an abelian group by U_{d} · k_{q−1}(F). In this section, we prove that a symbol in U^{d}k_{q}(F) can be written as {u} · α where u ∈ U_{d} and α is a symbol in k_{q−1}(F) (and not just as a sum of such things). More precisely, we have:

16.1. Gathering Lemma. For every nonzero symbol γ ∈ k_{q}(F) with q ≥ 2, there is a u ∈ 𝔬^{×} and a symbol α ∈ k_{q−1}(F) such that γ = {u} · α, depth{u} = depth γ, and depth α = 0. The symbol α may be chosen to be {a_{2}, …, a_{q}} with 0 ≤ λ(a_{2}) < p and λ(a_{i}) = 0 for 3 ≤ i ≤ q. If depth γ is not divisible by p, we may further arrange that λ(a_{2}) has any pre-assigned value j = 0, …, p − 1 as desired.

One should compare this lemma in the case p = 2 and q = 2, 3 with the Normal Form Theorems 11.11 and 11.12. Heuristically speaking, here we “gather the depth in the first slot” u. The Normal Form Theorems (in view of Th. 19.2 below and with u replaced by L) do the same, except when C is of unitary type, where they take depth L = depth C − 1.

We first amass some preliminary results; the proof of the Gathering Lemma will come at the end of the section. We use only background material on K-theory including the material summarized in §14; we don’t use anything else from this paper.

16.2. We write O(π^{i}) for an unspecified element (possibly zero) of 𝔬 divisible by π^{i}. We have the trivial but useful observation:

u + O(π^{j}) = u(1 + O(π^{j})) for u ∈ 𝔬^{×} and j ≥ 0.

Indeed, for b ∈ 𝔬, we have: u + bπ^{j} = u (1 + (u^{−1}b)π^{j}).

16.3. Example (cf. [4, p. 122]). Suppose that y ∈ 𝔬 has ȳ = c̄^{p} for some c ∈ 𝔬 and fix 0 ≤ s < λ(p)/(p − 1) such that 1 + yπ^{ps} ∈ 𝔬^{×}. Then

1 + yπ^{ps} = 1 + c^{p}π^{ps} + O(π^{ps+1}) = (1 + cπ^{s})^{p} + O(π^{ps+1})

where the second equality is because

(1)  λ (π^{is}\binom{p}{i}) ≥ is + λ(p) > is + (p − 1)s ≥ ps for 1 ≤ i < p.

As 1 + yπ^{ps} is a unit, so is (1 + cπ^{s})^{p}, and 16.2 gives that 1 + yπ^{ps} = (1 + cπ^{s})^{p} (1 + O(π^{ps+1})), hence {1+yπ^{ps}} = {1+O(π^{ps+1})} in k_{1}(F). Noting that the hypotheses on y obviously depend only on ȳ (or applying 16.2 once more), we find:

{1 + yπ^{ps} + O(π^{ps+1})} = {1 + O(π^{ps+1})} in k_{1}(F).

16.4. Lemma. Let a, b ∈ 𝔬, i ≥ 0, j ≥ 1 with 1 + aπ^{i} ∈ 𝔬^{×}. Then in k_{q}(F) we have:

{1 + aπ^{i}, 1 + bπ^{j}} = {1 + cπ^{i+j}, dπ^{i(p−1)}}

for some nonzero c, d ∈ 𝔬. Further, if a, b ∈ 𝔬^{×}, then also c, d ∈ 𝔬^{×}.

Proof. The computations in the proof of [10, 4.1.1b] or [4, p. 122] yield:

\{1 + aπ^{i}, 1 + bπ^{j}\} = − \{1 + \frac{ab}{1 + aπ^{i}} π^{i+j}, −aπ^{i}(1 + bπ^{j})\} ∈ k_{2}(F).

As −1 = p − 1 in k_{0}(F), we have:

\{1 + aπ^{i}, 1 + bπ^{j}\} = \{1 + \frac{ab}{1 + aπ^{i}} π^{i+j}, (−a(1 + bπ^{j}))^{p−1} π^{i(p−1)}\} .  □

16.5. Lemma. Suppose that {1+bπ^{ps}, a_{2}, …, a_{q}} satisfies db̄∧dā_{2}∧dā_{3}∧⋯∧dā_{q} = 0 in Ω^{q}, with b, 1 + bπ^{ps}, a_{2}, …, a_{q} ∈ 𝔬^{×} and 0 ≤ s < λ(p)/(p − 1). Then {1 + bπ^{ps}, a_{2}, …, a_{q}} is equal to {u′, a′_{2}, …, a′_{q}} for some u′, a′_{i} ∈ 𝔬^{×} where depth{u′} > ps.

Proof. We may assume that db̄ is not zero—hence that b̄ is not a p-th power—by Example 16.3. This settles the q = 1 case.

Suppose q ≥ 2 and \{ā_{2}, …, ā_{q}\} is zero in k_{q−1}(\overline{F}), so dā_{2} ∧⋯∧dā_{q} = 0 by 14.3. We apply the q − 1 case of the lemma with u = a_{q} and s = 0 (so u = a_{q} = 1 + c with c = a_{q} − 1, hence dc̄ = dā_{q}) to see that {a_{2}, …, a_{q}} = {a′_{2}, …, a′_{q}} where depth{a′_{q}} is positive. Then Lemma 16.4 gives the claim.

So we may assume that {ā_{2}, …, ā_{q}} is not zero. Write [i] for a (q − 1)-tuple (i_{2}, …, i_{q}) with 0 ≤ i_{j} < p and put ā^{[i]} for ā_{2}^{i_{2}}ā_{3}^{i_{3}} ⋯ ā_{q}^{i_{q}} ∈ \overline{F}. By 14.3 there are c_{[i]} ∈ 𝔬 such that

b̄ = (∑_{[i]} c̄_{[i]} \sqrt[p]{ā^{[i]}})^{p} = ∑_{[i]} c̄_{[i]}^{p}ā^{[i]}.

If it happens that c̄_{[i]} = 0 for all nonzero [i], then b̄ is a p-th power in \overline{F} and we are done. We now show, roughly speaking, that we can make c̄_{[i]} zero for all nonzero [i].

More precisely, fix a nonzero [i] with c̄_{[i]} nonzero; choose a specific j_{0} such that i_{j_{0}} ≠ 0. Take E/F to be the extension obtained by adjoining a p-th root α of a_{j_{0}}^{i_{j_{0}}} ∏_{j≠j_{0}}(−a_{j})^{i_{j}}; obviously α is integral. Take v := c_{[i]}α. As c̄_{[i]} is not zero, the residue of v does not belong to \overline{F}, hence vπ^{s} is not in F and so has minimal polynomial x^{p} − (c_{[i]}απ^{s})^{p} in F [x]. By degree count, this is also the characteristic polynomial chpoly_{vπ^{s}}(x) of vπ^{s} as an element of the F -algebra E. It follows that

(1)  N_{E/F} (1 − vπ^{s}) = chpoly_{vπ^{s}}(1) = 1 + (−1)^{p}N_{E/F} (v)π^{ps}.

Next observe that E kills γ := {a_{2}, …, a_{q}}: we renumber the a_{j}’s so that j_{0} = 2. In k_{q−1}(E), we have:

i_{2}γ = {a_{2}^{i_{2}}, a_{3}, …, a_{q}} = ∑_{j=3}^{q}(p − i_{j}){−a_{j}, a_{3}, …, a_{q}} = 0,

where the middle equality is because α is in E. As i_{2} is not divisible by p, we deduce that γ is zero in k_{q−1}(E), as required.

Now the projection formula [19, 7.2.7] gives:

0 = N_{E/F} ({1 − vπ^{s}} · γ) = {N_{E/F} (1 − vπ^{s})} · γ

in k_{q}(F). Combining this with (1), we find:

{1 + bπ^{ps}} · γ = {1 + (b + (−1)^{p}N_{E/F} (v))π^{ps} + O(π^{ps+1})} · γ.

But the residue of N_{E/F} (v) is v̄^{p} = c̄_{[i]}^{p}ā^{[i]}. In this way, we have replaced b with a new one that has fewer nonzero coefficients c̄_{[i]}. Repeating this process completes the proof. □

Proof of the Gathering Lemma 16.1. First, consider a symbol {x, y} ∈ k_{2}(F). Suppose that neither λ(x) nor λ(y) are divisible by p. Then there is some s such that λ(x) ≡ sλ(y) (mod p) and {x, y} = {x(−y)^{−s}, y} because {−y, y} is zero in k_{2}(F). As λ(x(−y)^{−s}) = λ(x) − sλ(y), we may assume that x has value 0.

Second, we may shuffle the entries in a symbol {x_{1}, …, x_{q}} by a permutation σ. We have {x_{1}, …, x_{q−1}, x_{q}} = {x_{σ(1)}, …, x_{σ(q−1)}, x_{σ(q)}^{sgn σ}}.

Combining the two preceding paragraphs shows that we may write γ = {u} · α for α = {a_{2}, …, a_{q}} with u, a_{i} ∈ 𝔬^{×} for 3 ≤ i ≤ q. Amongst all such ways of writing γ, fix one with depth{u} maximal. For sake of contradiction, suppose that d := depth{u} < depth γ ≤ m. Put r = 2 if λ(a_{2}) = 0 and r = 3 if 0 < λ(a_{2}) < p.

We now inspect the Kato isomorphism at depth d. By hypothesis, γ is zero in gr^{d} k_{q}(F), so has zero image. If d = 0, then \{ū, ā_{r}, …, ā_{q}\} is zero in k_{∗}(\overline{F}), hence d(ū − 1\overline{F}) ∧ dā_{r} ∧ ⋯ ∧ dā_{q} is zero by 14.3. Lemma 16.5 gives a contradiction. The case where d = ps for some 0 < s = d/p < (depth γ)/p < λ(p)/(p − 1) is similar.

Finally we suppose d is not divisible by p. We claim that λ(a_{2}) may be freely chosen. Indeed, for j ∈ ℤ, λ(a_{2}) − j ≡ sd (mod p) for some s ≥ 0. Writing u = 1 + bπ^{d} for b ∈ 𝔬^{×}, we have:

{1 + bπ^{d}, a_{2}} = {1 + bπ^{d}, a_{2}} − s{1 + bπ^{d}, −bπ^{d}} = {1 + bπ^{d}, a_{2}(−b)^{−s}π^{−sd}},

where λ(a_{2}(−b)^{−s}π^{−sd}) ≡ j (mod p), proving the claim. The hypothesis that {u}·α has depth greater than d implies that dā_{2} ∧ ⋯ ∧ dā_{q} is zero, and applying Lemma 16.5 to α with s = 0 shows that we may assume that one of the a_{i} has residue 1; Lemma 16.4 gives a contradiction. □

16.6. Remark. We can quickly deduce a useful restatement of the Gathering Lemma. Fix some r ≥ 1 and a permutation σ of {2, …, q} and put

β := {u, a_{σ(2)}, …, a_{σ(r)}} and δ := {a_{σ(r+1)}, a_{σ(r+2)}, …, a_{σ(q)}}.

Because of the identity {x, y} = −{y, x} in K_{2}^{M}(F), we find:

γ = ±β · δ, depth β = depth γ, and depth δ = 0.

Indeed, for the second and third equalities, ≥ is obvious and ≤ follows from the first equality and equation (14.1.1). We will apply this in the case p = 2, so the sign in the first equation will be irrelevant.

17. Ramification index for symbols

17.1. Definition. For a class γ ∈ k_{q}(F), we put e_{γ} = p if • depth(γ) is not divisible by p, or

• depth(γ) is divisible by p and its initial form has nonzero projection in the second summand of gr^{d} k_{q}(F). (Note that this condition does not depend on the choice of uniformizer π.)

Otherwise—or if γ = 0—we put e_{γ} = 1.

The “ramification index” e_{γ} is more subtle than in the case of good residue characteristic, see Example 20.4 below. But we do have the following positive results:

17.2. Proposition. Let γ ∈ k_{q}(F), q ≥ 2, be a non-zero symbol. Then there exist a symbol β ∈ k_{q−1}(F) and an element a ∈ F^{×} such that γ = β · {a}, e_{β} = 1 and one of the following holds.

(i) depth β = 0, depth{a} = depth γ, a ∈ 𝔬^{×} and
e_{γ} = 1 ⟺ depth γ ≡ 0 mod p.
(ii) depth β = depth γ, 0 < λ(a) < p and e_{γ} = p.

Proof. Write γ as in the Gathering Lemma 16.1, where we may assume a_{2}, …, a_{q−1} ∈ 𝔬^{×}, 0 ≤ λ(a_{q}) < p. If λ(a_{q}) = 0, then the Kato isomorphism at depth zero shows e_{α} = 1, so with β := α and a = u^{±1} we are in Case (i) since the Kato isomorphism at depth d := depth γ gives e_{γ} = 1 iff d is divisible by p. Now suppose λ(a_{q}) > 0; by the final statement of the Gathering Lemma, we may also assume d ≡ 0 mod p. Arguing as before, in particular consulting the Kato isomorphisms again for the determination of e_{γ}, β := {u, a_{2}, …, a_{q−1}} has depth d and e_{β} = 1, so we are in Case (ii). □

17.3. Proposition. For a symbol γ ∈ k_{q}(F) for q ≥ 2, the following are equivalent:

(i) γ = β · {a} for a nonzero symbol β ∈ k_{q−1}(F) with e_{β} = 1 and a ∈ F^{×} of value not divisible by p.

(ii) e_{γ} = p and depth γ is divisible by p. If these conditions hold, then additionally depth γ = depth β.

Proof. The proof amounts to looking at the explicit formulas for the Kato isomorphisms.

\underline{(i) ⇒ (ii)}: The Kato isomorphisms send β to a nonzero symbol β̄ (in some cohomology group over \overline{F}), because e_{β} = 1. We have depth(γ) ≥ depth(β) by (14.1.1), and an examination of the isomorphisms show that the isomorphism at the depth of β (divisible by p) sends β · {a} to λ(a)β̄ in the second component of the image, which is not zero because all of the targets of the Kato isomorphisms are abelian groups killed by p; that is, e_{γ} = p and depth γ = depth β.

\underline{(ii) ⇒ (i)}: By (ii), alternative (i) of Prop. 17.2 does not hold. Hence alternative (ii) does. □

18. Theorem 9.9 revisited

We will now prove a version of Theorem 9.9 for K-theory mod-2 when \overline{F} has characteristic 2; in the notation of §14 we restrict to the case p = 2. We replace the anisotropic round quadratic form P with e_{P/F} = 1 from Th. 9.9 with a nonzero symbol γ ∈ k_{q}(F) with q ≥ 1 and e_{γ} = 1 (in particular, depth γ is even). We replace the hypothesis on texp with the hypothesis depth{μ} + depth γ = m (recall that m = 2λ(2)), the largest possible depth for a nonzero element of k_{q}(F). The extreme cases where depth γ is 0 or m are comparatively easy, so we focus on the middle case. We will use the following:

18.1. Technique. If one has a nonzero class b̄\frac{dā_{2}}{ā_{2}} ∧ ⋯ ∧ \frac{dā_{q}}{ā_{q}} ∈ Ω^{q−1}/Z^{q−1}, we

may apply d and obtain the nonzero symbol x_{0} := db̄ ∧ \frac{dā_{2}}{ā_{2}} ∧ ⋯ ∧ \frac{dā_{q}}{ā_{q}} in Ω^{q}. The \overline{F} -span of this symbol contains x := b̄^{−1}x_{0}, which lies in ν(q). Indeed, it is the unique nonzero element of \overline{F} x_{0} with this property, because ν(q) is defined to be ker(γ − 1) for γ the inverse Cartier operator [10, pp. 123, 124] and for c ∈ \overline{F} we have (γ − 1)(cx) = (c^{2} − c)x ∈ Ω^{q}. A canonical isomorphism identifies x with the class of the anisotropic symmetric bilinear form B = ⟨⟨b̄, ā_{2}, …, ā_{q}⟩⟩ in the graded Witt ring [26], hence with B itself [15, 6.20]. The equivalence from Prop. 3.3 takes B and gives an extension K/\overline{F} of degree 2^{q} with a unital linear form s: K → \overline{F}.

Let γ ∈ k_{q}(F) be a nonzero symbol of even depth d ≠ 0, m and suppose that e_{γ} = 1. Then the initial form of γ is a nonzero symbol in Ω^{q−1}/Z^{q−1} and the technique in the preceding paragraph gives a (K, s) derived from γ.

18.2. Proposition. Let γ ∈ k_{q}(F) be a nonzero symbol of even depth d ≠ 0, m with e_{γ} = 1. Write (K, s) as in 18.1, and write γ = {1 + xπ^{d}} · α with x ∈ 𝔬^{×} as in the Gathering Lemma 16.1. For μ = 1 − bπ^{m−d} with b ∈ 𝔬^{×}, we have: {μ} · γ = 0 in k_{q+1}(F) if and only if the residue of xbπ^{m}/4 is in the image of ℘_{K,s}.

Proof. Write α = {a_{2}, …, a_{q}} for some a_{2}, …, a_{q} ∈ 𝔬^{×}. Using Lemma 16.4, we calculate:

\{μ\} · \{1 + xπ^{d}\} = \{1 + xπ^{d}, 1 − bπ^{m−d}\} = \{1 − \frac{bx}{1 + xπ^{d}} π^{m}, −x(1 − bπ^{m−d})\} .

We see from this that the initial form of \{μ\} · γ is x̄b̄ε̄\frac{dx̄}{x̄} ∧ \frac{dā_{2}}{ā_{2}} ∧ ⋯ ∧ \frac{dā_{q}}{ā_{q}} where we have set ε := π^{m}/4 to simplify the notation. This determines the class of the quadratic Pfister form ⟨⟨x̄, ā_{2}, …, ā_{q}, x̄b̄ε̄⟧ in the graded Witt group of quadratic forms [26]. The Arason-Pfister Hauptsatz [15, 23.7(1)] implies that this class is zero (equivalently, {μ} · γ is zero) if and only if the Pfister form is isotropic, if and only if x̄b̄ε̄ is in the image of ℘_{K,s} by Cor. 3.10(a), proving the claim. □

19. Dictionary between K-theory and algebras and quadratic forms

In the cases q = 1, q = 2, or p = 2, we have a close relationship between properties of symbols in k_{q}(F) relative to Kato’s filtration and valuation-theoretic properties on the corresponding algebras. Specifically:

19.1. Proposition. In cases q = 1 and q = 2 we have:

(i) e_{γ} = e_{D/F}.

(ii) \overline{D} is a separable division algebra over \overline{F} and distinct from \overline{F} if and only if depth(γ) = m.

That is, you can determine e_{D/F} and whether or not D is tame by examining the corresponding symbol in k_{q}(F).

We prove the proposition in §20 and 21.2 below. The calculations appearing in the proof of this result are similar to some in sections 1.1 and 2.1 of [52]. Roughly speaking, our statements here differ from those in [52] by using the language of Kato’s filtration.

In contrast with Proposition 19.1, the following theorem relies heavily on the results of Part II. It is proved in §22.

19.2. Theorem. Let p = 2. Fix a nonzero symbol γ ∈ k_{q}(F) for some q ≥ 1 and let Q be the corresponding anisotropic q-Pfister form. Then:

(i) e_{γ} = e_{Q/F}.

(ii) \overline{Q} is nonsingular (i.e., Q is tame) if and only if depth(γ) = 2λ(2).

(iii) texp(Q) = λ(2) − ⌊\frac{depth(γ)}{2}⌋.

In the statement of the theorem, Q is a quadratic form, whereas the results in §§7–9 (including the definition of tame and texp) are in terms of pointed quadratic spaces. This is harmless in view of Prop. 7.3. The theorem leads to:

19.3. Corollary. Fix a division quaternion or octonion algebra C over F and write γ for the corresponding symbol in k_{∗}(F). We have:

(i) C is of primary type if and only if e_{γ} = 2 and depth(γ) is even. (Otherwise C is of unitary type.)

(ii) ω(C) = λ(2) − \frac{depth(γ)}{2}

That is, one can read off properties of the composition algebra C—including the invariant ω(C) studied by Saltman and Tignol—from the properties of the corresponding symbol γ in Milnor K-theory. We prove Theorem 19.2 and Cor. 19.3 in §22.

19.4. Remark. It is natural to wonder if on can extend Prop. 19.1 to include the case p = q = 3, where the corresponding algebraic objects are Albert (Jordan) algebras obtained from the first Tits construction. The answer is no, because we do not know if two such Albert algebras corresponding to the same symbol in k_{3}(F) are necessarily isomorphic. (This is a special case of open problem #4 from [42].) If this is indeed the case, then one can easily extend Prop. 19.1 to include the case p = q = 3 by taking advantage of the valuation theoretic results in [40] and by imitating the proof in the cases given below.

20. Proof of Proposition 19.1: case q = 1

20.1. Example (q = 1 and nonzero depth divisible by p). Fix {x} ∈ k_{1}(F) of depth d divisible by p, so x = 1 + uπ^{d} for some u of value 0. As the “second summand” Ω^{−1}/Z^{−1} or H^{1}(\overline{F} , ν(−1)) in the Kato isomorphism is zero, e_{\{x\}} = 1.

The algebra D corresponding to {x} is F (χ) where χ^{p} = x. For

α := π^{−d/p}(χ − 1) ∈ D

we have

(α + π^{−d/p})^{p} = π^{−d}x = u + π^{−d},

so

(1)  ∑_{i=1}^{p} \binom{p}{i}α^{i}π^{−(p−i)d/p} − u = 0.

For 1 ≤ i < p, the prime p divides \binom{p}{i}, so

λ (\binom{p}{i}π^{−(p−i)d/p}) ≥ λ(p) − \frac{d}{p}(p − i) = \frac{(m − d)(p − 1) + d(i − 1)}{p}.

As i ≥ 1 and d ≤ m, this is at least 0, hence α is integral. Further, the coefficient of α^{i} in (1) has residue zero for 2 ≤ i < p in all cases and also for i = 1 if d < m. (Clearly, ᾱ is not zero, so λ(α) = 0.)

Therefore, if d < m, \overline{D} contains ᾱ satisfying ᾱ^{p} − ū = 0. As x has depth d, the Kato isomorphism shows that ū is not a p-th power in \overline{F}, and we conclude that \overline{D} is the proper extension \overline{F} (\sqrt[p]{ū}) and e_{D/F} = 1.

In case d = m, we set η := ζ − 1. As λ(η) = m/p = λ(p)/(p − 1), we have

x = 1 + bη^{p} for b = \frac{uπ^{m}}{η^{p}} .

We put β := η^{−1}(χ − 1) and apply the same reasoning as in the case d < m with π^{−d/p} replaced with η^{−1}. The element β satisfies

(2)  ∑_{i=1}^{p} \binom{p}{i}β^{i}η^{−(p−i)} − b = 0.

Again, the coefficients of β are integral because

(3) λ (\binom{p}{i}η^{−(p−i)}) ≥ λ(p) − (p − i)\frac{λ(p)}{p − 1} = λ(p)\frac{i − 1}{p − 1} ≥ 0 for 1 ≤ i < p.

Taking residues of (2) kills the terms with 1 < i < p. For the i = 1 term, we note that expanding the equation (1 + η)^{p} = 1 gives ∑_{i=1}^{p} \binom{p}{i}η^{i} = 0, so pη = − ∑_{i=2}^{p} η^{i} and

pη^{−(p−1)} = pη^{1−p} = − ∑_{i=2}^{p} \binom{p}{i}η^{−(p−i)}.

Taking residues and applying (3), only the i = p term is nonzero on the right side, so pη^{−(p−1)} has residue −1. Taking residues of (2), we find the equation β̄^{p} − β̄ − b̄ = 0 in \overline{D}. The fact that x has depth d asserts that the element b̄ is nonzero in H^{1}(\overline{F} , ν(0)) ≅ \overline{F} /℘(\overline{F}), i.e., β̄ generates a proper extension of \overline{F} and we conclude that D/F is unramified. We have verified Proposition 19.1 for γ ∈ k_{1}(F) of nonzero depth divisible by p.

20.2. Remark. It might be illuminating to compare Example 20.1 in the case p = 2 with the material in Part II. In case 0 < d < m = 2λ(2) = 2 texp(F), we compare the example to Prop. 9.7 with P = F. The Kato isomorphism at depth d sends {x} to the image of ū in F̄/F̄^{2}, and we conclude ū ∉ F̄^{2}, in agreement with Prop. 9.7. Moreover, identifying F (\sqrt{x}) = Cay(F, x) = F ⊕ F j and j = −χ, we obtain that −α = Ξ in the sense of (9.7.2); it is a normalized trace generator.

The situation with d = 2λ(2) is slightly more delicate; we compare the example to Thm. 9.9 (resp. Cor. 10.18) for P (resp. C) = F. First of all, we have ζ = −1, η = −2, β = \frac{χ−1}{2}, and w_{0} = \frac{π^{λ(2)}}{2} ∈ 𝔬^{×} is the unique normalized trace generator of F. Moreover, s̄_{w_{0}} = 1F̄, forcing ℘_{F̄,s̄_{w_{0}}} = ℘ to be the usual Artin-Schreier map on F̄. We have u = −β, u_{0} = n_{F} (w_{0})u = −β_{0} in the sense of (9.9.1) and b = u_{0}. The Kato isomorphism at depth m sends {x} to ū_{0} = β̄_{0} ∈ H^{1}(F̄, ν(0)) = F̄/℘(F̄), forcing β̄_{0} ∉ Im(℘) in agreement with the equivalence (iii) ⇔ (v) in Thm. 9.9. Furthermore, by Cor. 10.18,

C̄ = F̄[t]/(t^{2} − t − ū_{0}) = F̄[t]/(t^{2} − t − b̄) = \overline{F (\sqrt{x})},

in agreement with the second part of Example 20.1.

20.3. Proof of Proposition 19.1 for q = 1: Fix a nonzero {x} ∈ k_{1}(F) and write D = F (χ) as in Example 20.1.

Suppose first that x has depth 0; we may assume that 0 ≤ λ(x) < p. The Kato isomorphism

gr^{0} k_{1}(F) ⟶^{∼} k_{1}(\overline{F}) ⊕ k_{0}(\overline{F}) ≅ \overline{F}^{×}/\overline{F}^{×p} ⊕ ℤ/pℤ

is the direct sum of the specialization map s_{π} and the tame symbol ∂, described concretely in [19], e.g. As ∂(x) ≡ λ(x) (mod p), we have e_{{x}} = p if and only if λ(x) is not zero. As to the extension D/F, the element χ satisfies χ^{p} − x = 0, hence is integral. If λ(x) is not zero, then χ̄^{p} is zero, hence χ̄ is zero, \overline{D} = \overline{F}, and e_{D/F} = p. If λ(x) is zero, then since x has depth 0, x̄ is not a p-th power in \overline{F} and \overline{F} (χ̄) is a proper extension of \overline{F}; in this case \overline{D} = \overline{F} (\sqrt[p]{x̄}) and e_{D/F} = 1.

Suppose that x has depth d not divisible by p; we may take x of the form 1+uπ^{d} where u has value 0. Then

N_{D/F} (χ − 1) = (−1)^{p−1}(x − 1) = (−1)^{p−1}uπ^{d},

so the value of χ − 1 is not an integer and e_{D/F} = p = e_{{x}}.

Note that in both of these cases, \overline{D} is not a proper separable extension of \overline{F}. The remaining case where d has nonzero depth divisible by p was treated in Example 20.1, which concludes the proof. □

20.4. Example. Let \overline{F} := 𝔽_{2}((x)), Laurent series over the field with 2 elements. Construct a field F of characteristic zero by taking the absolutely unramified Teichmüller extension of \overline{F} as in Example 10.15 and adjoining \sqrt{2}. Then F has residue field \overline{F} and \sqrt{2} is a uniformizer. Consider the elements a = 1 + \sqrt{2}^{3} and b = 1 + 2u in F, where u ∈ F is a unit with residue x. The symbols {a} and {b} have depth 3 and 2. The valuation λ ramifies on F (\sqrt{a}) and does not ramify on F (\sqrt{b}).

A reader familiar with the situation of good residue characteristic might assume that the quaternion algebra (a, b) is division over F, but in fact it is split. This is easily seen because the symbol {a, b} has depth at least 5 by (14.1.1), which is greater than 2λ(2).

21. Proof of Proposition 19.1: cases q = 2

We now prove Proposition 19.1 and Theorem 19.2 in the case q = 2. We consider a nonzero symbol {x, y} in k_{2}(F) and write D for the corresponding symbol algebra.

21.1. Lemma. e_{D/F} = p if and only if some subfield L of D has e_{L/F} = p.

Proof. Easy. If such an L exists, then there is some ℓ ∈ L^{×} such that λ(N_{L/F} (ℓ)) is not divisible by p. But N_{L/F} (ℓ) = Nrd_{D/F} (ℓ) by [11, p. 150]. This proves “if”. For “only if”, note that a nonzero element of D generates a subfield L/F. □

21.2. Proof of Proposition 19.1 for q = 2: Suppose first that the depth d of {x, y} is not divisible by p (so e_{{x,y}} = p).

By the Gathering Lemma 16.1 we can arrange that x is in U_{d} and y ∈ 𝔬^{×} has residue that is not a p-th power. As in 20.3, e_{F (χ)/F} = p, hence e_{D/F} = p and the dimension of \overline{D}/\overline{F} is p. The residue field of F (ψ) is the proper extension \overline{F} (\sqrt[p]{ȳ}) of \overline{F}, and by dimension count it is all of \overline{D} and D is wild. So we may assume that the depth of {x, y} is divisible by p.

Suppose now that e_{{x,y}} = p. By Prop. 17.3, we may assume that e_{{x}} = 1 and y = uπ^{n} for some u ∈ 𝔬 and n not divisible by p, and depth{x, y} = depth{x}. On the one hand, D contains F (ψ) on which the valuation ramifies, so e_{D/F} = p as claimed. On the other hand, D contains F (χ) whose residue algebra is a proper extension of \overline{F} that is inseparable (if depth\{x, y\} < m) or separable (if depth{x, y} = m) by Example 20.1; this proves the claim. We are left with the case where the depth is divisible by p and e_{{x,y}} = 1.

If the depth of {x, y} is zero, then by the Kato isomorphisms {x̄, ȳ} is not zero in k_{2}(\overline{F}), hence x̄, ȳ are p-free over \overline{F}^{p}. Following 20.3, χ̄ and ψ̄ generate purely inseparable extensions of \overline{F} in \overline{D}. The value of χψ − ψχ = (ζ − 1)ψχ is λ(ζ − 1) = m/p > 0, so χ̄ and ψ̄ commute in \overline{D}. We deduce that \overline{D} is \overline{F} (\sqrt[p]{x̄}, \sqrt[p]{ȳ}), e_{D/F} = 1, and D is wild.

If 0 < depth{x, y} < m, then we can choose x = 1 + aπ^{d} where d = depth{x, y} so that the initial form of \{x, y\} in Ω^{1}/Z^{1} is ā\frac{dȳ}{ȳ}. As the depth is d, ā\frac{dȳ}{ȳ} is not in Z^{1}, i.e., dā ∧ \frac{dȳ}{ȳ} is not 0. It follows that dim\overline{F} \overline{F} (\sqrt[p]{ā}, \sqrt[p]{ȳ}) = p^{2} by 14.3. We claim that this field is \overline{D}. By dimension count, it suffices to note that for α := π^{−d/p}(χ − 1), we have

ᾱ^{p} = ā, ψ̄^{p} = ȳ, and ᾱψ̄ = ψ̄ᾱ.

The first equation is as in Example 20.1 and the second is obvious. The third follows because

αψ − ψα = π^{−d/p}(χψ − ψχ) = π^{−d/p}(ζ − 1)ψχ.

But ζ − 1 has value m/p > d/p, hence this commutator has positive value and ᾱ, ψ̄ commute. This shows that e_{D/F} = 1 and D is wild.

Finally suppose that the depth of {x, y} is m. Then we write x = 1 + bη^{p} and β = η^{−1}(χ − 1) as in Example 20.1; again β̄^{p} − β̄ − b̄ = 0. Further,

βψ − ψβ = η^{−1}(χψ − ψχ) = ψχ,

so

β̄ψ̄ = ψ̄(β̄ + 1).

The elements β̄ and ψ̄ generate a division algebra of dimension p^{2} over its center \overline{F} [19, p. 36], so it must be \overline{D}. Therefore, e_{D/F} = 1 and D is tame. □

22. Proofs of Theorem 19.2 and Corollary 19.3

The following proofs amount to translating results of §§8, 9 into K-theory.

Proof of Theorem 19.2. It suffices to establish (i) and (iii) since by Prop. 8.2(a) (iii) implies (ii).

\underline{Case q = 1}: Suppose first that q = 1, i.e., γ = \{μ\} for some μ ∈ F^{×} and Q is the 1-Pfister ⟨⟨μ⟩⟩. Put P := ⟨1⟩; it is wild because char \overline{F} = 2 and texp P = λ(2) by Example 8.7. We may assume that μ has value 0 or 1. If μ has value 1, then depth γ = 0, e_{γ} = 2, and the theorem holds by Prop. 9.2.3. If μ has value 0 and μ̄ is a nonsquare in \overline{F}, then depth γ = 0, e_{γ} = 1, and the theorem holds by Prop. 9.7.6.

Otherwise \overline{μ} is a square in \overline{F}, so multiplying μ by a square in F we may assume that \overline{μ} = 1. If depth γ < 2λ(2), then (i) and (iii) hold by Prop. 9.6.5 or Prop. 9.7.6.

Finally suppose that depth γ = 2λ(2). The symbol γ corresponds to the quadratic extension F (\sqrt{μ}) and the residue algebra of this extension was computed in Example 20.1; this verifies (i) and that Q is tame (hence (ii)). Then texp Q = 0 by Prop. 8.2(d), proving (iii).

\underline{Case q ≥ 2}: We argue by induction on q and decompose γ as in Prop. 17.2 (with p = 2). Writing P (resp. Q) for the Pfister quadratic form corresponding to β (resp. γ), we conclude Q ≅ ⟨⟨a⟩⟩ ⊗ P and put d := depth γ. Then e_{P/F} = 1 and texp(P) = λ(2) − (depth β)/2 by the induction hypothesis. If alternative (ii) of Prop. 17.2 holds, then (9.2.3) shows e_{γ} = 2 = e_{Q/F} and texp(Q) = texp(P) =

λ(2) − ⌊\frac{depth γ}{2} ⌋ since depth γ = depth β is even. Now suppose alternative (i) of Prop. 17.2 holds. Since β has depth zero and ramification index 1, we can write β = {a_{1}, …, a_{q−1}}, a_{i} ∈ 𝔬^{×}, 1 ≤ i < q. If d = 2λ(2), then e_{γ} = 1 and ⟨⟨a⟩⟩ is a tame 1-Pfister quadratic subspace of Q ≅ ⟨⟨a, a_{1}, …, a_{q−1}⟩⟩ with e_{⟨⟨a⟩⟩/F} = 1. Applying Prop. 9.3 q − 1 times yields e_{Q/F} = 1 = e_{γ} and that Q is tame as well, forcing texp(Q) = 0 = λ(2) − ⌊d/2⌋. We are left with the case 0 ≤ d < 2λ(2). Assertions (i) and (iii) follow by combining the effect of the Kato isomorphism at depth d on γ with 14.3 and (9.6.5) (for d odd) or with (9.7.6) and Cor. 9.10(a) (for d even). □

Proof of Corollary 19.3. We now prove Corollary 19.3. Claim (i) amounts to reformulating Proposition 17.3. For (ii), one combines Theorem 19.2(iii) relating depth γ with texp C with the relation between texp C and ω(C) demonstrated in the proof of Theorem 12.11. □

Acknowledgements. The first author was partially supported by National Science Foundation grant no. DMS-0653502. Both authors thank Eric Brussel, Detlev Hoffmann, Daniel Krashen, Ottmar Loos, Pat Morandi, David Saltman, and Jean-Pierre Tignol for useful comments and suggestions. Special thanks are due to Richard Weiss, whose stimulating questions (related to his recent work [56] on affine buildings) inspired us to look for successively improved versions of the Local Norm Theorems 8.10 and 15.2.

References

[1] A. A. Albert, Absolute valued real algebras, Ann. of Math. (2) 48 (1947), 495–501. MR 0020550 (8,561d)

[2] , On nonassociative division algebras, Trans. Amer. Math. Soc. 72 (1952), 296–309. MR 0047027 (13,816d)

[3] E. Becker, Über eine Klasse flexibler quadratischer Divisionsalgebren, J. Reine Angew. Math. 256 (1972), 25–57. MR 0320099 (47 #8640)

[4] S. Bloch and K. Kato, p-adic étale cohomology, Publ. Math. IHES 63 (1986), 107–152.

[5] N. Bourbaki, Elements of mathematics. Algebra, Part I: Chapters 1-3, Hermann, Paris, 1974, Translated from the French.

[6] , Algebra II, Springer-Verlag, 1988.

[7] R. B. Brown, On generalized Cayley-Dickson algebras, Pacific J. Math. 20 (1967), 415–422. MR 0215891 (35 #6726)

[8] F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. (1972), no. 41, 5–251. MR 0327923 (48 #6265)

[9] , Schémas en groupes et immeubles des groupes classiques sur un corps local, Bull. Soc. Math. France 112 (1984), no. 2, 259–301. MR 788969 (86i:20064)

[10] J.-L. Colliot-Thélène, Cohomologie galoisienne des corps valués discrets henseliens, d’après K. Kato et S. Bloch, Algebraic K-theory and its applications (Trieste, 1997) (H. Bass, A.O. Kuku, and C. Pedrini, eds.), 1999, pp. 120–163.

[11] P. K. Draxl, Skew fields, London Math. Soc. Lecture Note Series, vol. 81, Cambridge University Press, Cambridge-New York, 1983.

[12] A. Dress, Metrische Ebenen über quadratisch perfekten Körpern, Math. Z. 92 (1966), 19–29. MR 0206804 (34 #6620)

[13] H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel, and R. Remmert, Numbers, Graduate Texts in Mathematics, vol. 123, Springer, 1991.

[14] I. Efrat, Valuations, orderings, and Milnor K-theory, Mathematical Surveys and Monographs, vol. 124, American Mathematical Society, Providence, RI, 2006. MR 2215492 (2007g:12006)

[15] R. Elman, N. Karpenko, and A. Merkurjev, The algebraic and geometric theory of quadratic forms, American Mathematical Society Colloquium Publications, vol. 56, American Mathematical Society, Providence, RI, 2008. MR 2427530

[16] M.A. Elomary and J.-P. Tignol, Springer’s theorem for tame quadratic forms over Henselian fields, preprint, 2010.

[17] A. J. Engler and A. Prestel, Valued fields, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. MR 2183496 (2007a:12005)

[18] S. Garibaldi, A. Merkurjev, and J-P. Serre, Cohomological invariants in Galois cohomology, University Lecture Series, vol. 28, American Mathematical Society, Providence, RI, 2003. MR 1999383 (2004f:11034)

[19] P. Gille and T. Szamuely, Central simple algebras and Galois cohomology, Cambridge studies in Advanced Math., vol. 101, Cambridge, 2006.

[20] D.W. Hoffmann, Diagonal forms of degree p in characteristic p, Contemporary Math. 344 (2004), 135–183.

[21] D.W. Hoffmann and A. Laghribi, Quadratic forms and Pfister neighbors in characteristic 2, Trans. Amer. Math. Soc. 356 (2004), 4019–4053.

[22] B. Jacob, Quadratic forms over dyadic valued fields. I. The graded Witt ring, Pacific J. Math. 126 (1987), no. 1, 21–79. MR 868606 (87m:11039)

[23] K. Kato, A generalization of local class field theory by using K-groups. I, J. Fac. Sci. Univ. Tokyo IA 26 (1979), 303–376.

[24] , A generalization of local class field theory by using K-groups. II, J. Fac. Sci. Univ. Tokyo IA 27 (1980), 603–683.

[25] , Galois cohomology of complete discrete valuation fields, Lecture Notes in Math., vol. 967, pp. 215–238, Springer, 1982.

[26] , Symmetric bilinear forms, quadratic forms and Milnor K-theory in characteristic two, Invent. Math. 66 (1982), no. 3, 493–510.

[27] M. Knebusch, Generic splitting of quadratic forms. I, Proc. London Math. Soc. (3) 33 (1976), no. 1, 65–93. MR 0412101 (54 #230)

[28] , Specialization of quadratic and symmetric bilinear forms, http://www-nw.

uni-regensburg.de/~.knm22087.mathematik.uni-regensburg.de/book.pdf, 2007.

[29] M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The book of involutions, American Mathematical Society Colloquium Publications, vol. 44, American Mathematical Society, Providence, RI, 1998, With a preface in French by J. Tits. MR 2000a:16031

[30] T.Y. Lam, Introduction to quadratic forms over fields, Graduate Studies in Mathematics, vol. 67, Amer. Math. Soc., Providence, RI, 2005.

[31] O. Loos, The scheme of quaternionic algebras, To appear.

[32] P. J. McCarthy, Algebraic extensions of fields, second ed., Dover Publications Inc., New York, 1991. MR MR1105534 (92a:11138)

[33] K. McCrimmon, Nonassociative algebras with scalar involution, Pacific J. Math. 116 (1985), no. 1, 85–109.

[34] A.S. Merkurjev and A.A. Suslin, K-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Math. USSR Izv. 21 (1983), no. 2, 307–340.

[35] O. T. O’Meara, Introduction to quadratic forms, Grundlehren der mathematischen Wissenschaften, Vol. 117, Springer-Verlag, New York, 1963.

[36] J. M. Osborn, Quadratic division algebras, Trans. Amer. Math. Soc. 105 (1962), 202–221. MR 0140550 (25 #3968)

[37] Á. Rodríguez Palacios, Absolute-valued algebras, and absolute-valuable Banach spaces, Advanced courses of mathematical analysis I, World Sci. Publ., Hackensack, NJ, 2004, pp. 99– 155. MR 2162414 (2006j:46013)

[38] H. P. Petersson, Borel subalgebras of alternative and Jordan algebras, J. Algebra 16 (1970), 541–560. MR 52 #8209

[39] , Eine Bemerkung zu quadratischen Divisionsalgebren, Arch. Math. (Basel) 22 (1971), 59–61. MR 46 #9132

[40] , Jordan-Divisionsalgebren und Bewertungen, Math. Ann. 202 (1973MR50:422), 215– 243. MR 50 #422

[41] , Composition algebras over a field with a discrete valuation, J. Algebra 29 (1974), 414–426.

[42] H.P. Petersson and M.L. Racine, Albert algebras, Jordan algebras (Berlin) (W. Kaup, K. Mc-Crimmon, and H.P. Petersson, eds.), de Gruyter, 1994, (Proceedings of a conference at Oberwolfach, 1992), pp. 197–207.

[43] S. Pumplün, A non-orthogonal Cayley-Dickson doubling, J. Algebra Appl. 5 (2006), no. 2, 193–199. MR 2223466 (2007a:17009)

[44] F. Rosemeier, Semiquadratische algebren, Ph.D. thesis, FernUniversität in Hagen, 2002.

[45] D. J. Saltman, Division algebras over discrete valued fields, Comm. Algebra 8 (1980), no. 18, 1749–1774. MR MR589083 (83b:16015)

[46] T. A. Springer, Quadratic forms over fields with a discrete valuation. I. Equivalence classes of definite forms, Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math. 17 (1955), 352–362. MR 0070664 (17,17a)

[47] T. A. Springer and F. D. Veldkamp, Octonions, Jordan algebras and exceptional groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000.

[48] A. Suslin and S. Joukhovitski, Norm varieties, J. Pure Appl. Algebra 206 (2006), no. 1-2, 245–276.

[49] O. Teichmüller, Diskret bewertete perfekte Körper mit unvollkommenem Restklassenkörper, J. Reine Angew. Math. 176 (1937), 141–152.

[50] U.P. Tietze, Zur Theorie quadratischer Formen über Hensel-Korpern, Arch. Math. 17 (1955), 352–362.

[51] J.-P. Tignol, Algèbres à division et extensions de corps sauvagement ramifiées de degré premier, J. Reine Angew. Math. 404 (1990), 1–38.

[52] , Classification of wild cyclic field extensions and division algebras of prime degree over a Henselian field, Proceedings of the International Conference on Algebra, Part 2 (Novosibirsk, 1989) (Providence, RI), Contemp. Math., vol. 131, Amer. Math. Soc., 1992, pp. 491–508.

[53] F. van der Blij and T. A. Springer, The arithmetics of octaves and of the group G_{2}, Nederl. Akad. Wetensch. Proc. Ser. A 62 = Indag. Math. 21 (1959), 406–418. MR 0152555 (27 #2533)

[54] A.R. Wadsworth, Valuation theory on finite dimensional division algebras, Valuation theory and its applications, Vol. I (Saskatoon, SK, 1999), Fields Inst. Commun., vol. 32, Amer. Math. Soc., Providence, RI, 2002, pp. 385–449.

[55] R. M. Weiss, Quadrangular algebras, Mathematical Notes, vol. 46, Princeton University Press, Princeton, NJ, 2006. MR 2177056 (2007a:17010)

[56] , On the existence of certain affine buildings of type E_{6} and E_{7}, J. Reine Angew. Math. (To appear).

[57] K. A. Zhevlakov, A. M. Slin′ko, I. P. Shestakov, and A. I. Shirshov, Rings that are nearly associative, Pure and Applied Mathematics, Vol. 104, Academic Press, New York, 1982.

(Garibaldi) Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA

E-mail address: skip@mathcs.emory.edu

(Petersson) Fakultät für Mathematik und Informatik, FernUniversität in Hagen, D-58084 Hagen, Germany

E-mail address: holger.petersson@fernuni-hagen.de