DIMENSIONS OF ANISOTROPIC INDEFINITE QUADRATIC FORMS II

DETLEV W. HOFFMANN

Abstract. Let F be a field of characteristic different from 2. The u-invariant and the Hasse number ũ of a field F are classical and important field invariants pertaining to quadratic forms. These invariants measure the suprema of dimensions of anisotropic forms over F that satisfy certain additional properties. We prove new relations between these invariants and we give a new characterization of fields with finite Hasse number, the first one of its kind that uses intrinsic properties of quadratic forms and which, conjecturally, allows an ‘algebro-geometric’ characterization of fields with finite Hasse number. We also construct various examples of fields with infinite Hasse number and prescribed finite values of u that satisfy additional properties pertaining to the space of orderings of the field.

1. Introduction

Throughout this paper, fields are assumed to be of characteristic different from 2 and quadratic forms over a field are always assumed to be finite-dimensional and nondegenerate. The u-invariant of a field F is one of the most important field invariants pertaining to quadratic forms. Originally, it was defined to be the supremum of the dimensions of anisotropic quadratic form over F. As such, it doesn’t yield any useful information in the case of formally real fields (or real fields for short) as in that case, the quadratic form given by a sum of an arbitrary finite number of squares is always anisotropic. To provide a meaningful invariant also in the case of real fields, Elman and Lam [EL2] modified the definition as follows:

u(F) := sup{dim φ | φ is an anisotropic torsion form over F } ,

where ‘torsion’ means torsion when considered as an element in the Witt ring W F. Over a nonreal field, all forms are torsion forms, whereas over a real field, Pfister’s Local-Global Principle implies that torsion forms are exactly those forms that have zero signature with respect to every ordering of the field (see, e.g., [L3, Ch. VIII, Th. 3.2]). If F is real and there are no anisotropic torsion forms, one puts u(F) = 0 (for instance, for F = ℝ).

2000 Mathematics Subject Classification. primary: 11E04; secondary: 11E10, 11E81, 14C25.

Key words and phrases. quadratic form, Pfister form, Pfister neighbor, real field, ordering, strong approximation property, effective diagonalization, u-invariant, Hasse number, Pythagoras number, function field of a quadratic form, Rost correspondence, Rost projector.

There is a vast body of literature on the question of which values can be realized as u-invariant of a field, and on the problem of determining the u-invariant of certain fields such as finitely generated extensions of local or global fields. Despite much progress in recent years, many of these questions remain unanswered. For example, in the case where K is a finitely generated extension of ℚ_{p} of transcendence degree m ≥ 1, the finiteness of u(K) was not known until only recently, when Leep showed (based on results by Heath-Brown [He]) that in this situation one has u(K) = 2^{m+2}. Before that, this was only known in the case m = 1 and p > 2 due to Parimala-Suresh [PaS].

In the case where K is real and finitely generated over ℝ of transcendence degree m, it was Witt who showed that in the case m = 1, one has u(K) = 2. For m ≥ 2 the best currently known estimates are 2^{m} ≤ u(K) ≤ 2^{m+2} − 2m − 6 due to Becher [B2] (for m = 2, this is due to Elman-Lam [EL2]).

As for possible values of the u-invariant, it is not too hard to show that u cannot take the values 3, 5, 7 (see, e.g., [L3, Ch. XI, Prop. 6.8]). It was Merkurjev [M2] who showed that every even n ∈ ℕ can be realized as u-invariant of a field, thus also producing the first values different from 2-powers (see also [T]). Izhboldin [I] produced the first field with u = 9, and examples of fields with u = 2^{r} + 1 for all r ≥ 3 were constructed by Vishik [V]. It is still not known whether other values than the above ones are possible or not.

If F is a real field, for a form φ over F to be isotropic, it is clearly necessary for φ to be indefinite at each ordering of F, i.e., for φ to be totally indefinite or t.i. for short. This leads to another field invariant, the Hasse number ũ defined as

ũ(F) := sup{dim φ | φ is an anisotropic t.i. form} .

One puts ũ(F) = 0 if there are no anisotropic t.i. forms over F. Clearly, u(F) ≤ ũ(F), with equality in the case of nonreal fields since being totally indefinite is then an empty condition.

In [H6], u(F) and ũ(F) were related to another field invariant, the so-called symbol length λ(F), which is defined to be the smallest integer n such that each element in the exponent-2-part _{2}Br(F) of the Brauer group is Brauer equivalent to a product of ≤ n quaternion algebras. (By Merkurjev’s theorem [M1], each element in _{2}Br(F) is equivalent to a product of quaternion algebras.)

In the present paper, we focus on finiteness criteria for u and ũ. In particular, we derive a criterion for the finiteness of ũ that is of a purely quadratic form intrinsic nature, the first one of its kind, and which conjecturally can be translated into a ‘algebro-geometric’ criterion for the finiteness of ũ.

To formulate the result, we need to introduce two properties. The first one is for a field to have effective diagonalization, ED for short, see §2 for the definition. The second property is the Pfister neighbor property P N(n) for n ∈ ℕ. A field F having property P N(n) means that each form of dimension 2^{n} +1 over F is a Pfister neighbor (see §2 for the definition of Pfister neighbor).

The main result of this paper is the following. Theorem 1.1. Let F be field. Then the following statements are equivalent:

(i) ũ(F) < ∞.

(ii) u(F) < ∞ and F is ED.

(iii) F has property P N(n) for some n ≥ 2.

The equivalence (i)⟺(ii) is originally due to Elman-Prestel [EP], but we will give a new and simple proof that at the same time allows to improve considerably all previously known estimates for ũ in terms of u for ED-fields.

The paper is structured as follows. In §2, we recall some basic definitions and notations. In §3, we relate the property ED to the strong approximation property SAP and a certain property S_{1} concerning torsion binary forms. We give a new proof of the fact that ED is equivalent to SAP plus S_{1}, originally due to Prestel-Ware [PW]. In §4, we prove the equivalence (i)⟺(ii) in the above theorem, using a method that allows us to derive various estimates for ũ in terms of u. In §5, we construct examples of fields with prescribed values for u and the Pythagoras number p and which are SAP but not S_{1}, and which are S_{1} but not SAP, respectively (such fields will thus have infinite ũ). In §6, we prove the equivalence (i)⟺(iii), giving in fact estimates on ũ in terms of n if the field has property P N(n), n ≥ 2. Since property P N(2) is equivalent to F being linked (see Proposition 6.3), we will thus also recover as corollary a famous result on the u-invariant and the Hasse number of linked fields due to Elman-Lam [EL3], [E].

2. Definitions and notations

For all undefined terminology and basic facts about quadratic forms we refer to [L3]. Let φ and ψ be quadratic forms over a field F. We denote isometry, orthogonal sum and tensor product of φ and ψ by φ ≅ ψ, φ ⊥ ψ and φ ⊗ ψ, respectively. Since all fields are assumend to be of characteristic not 2 and all quadratic forms are assumed to be nondegenerate, any quadratic form φ over F can be diagonalized: φ ≅ ⟨a_{1}, …, a_{n}⟩ for some a_{i} ∈ F^{∗}.

φ is called a subform of ψ, φ ⊂ ψ, if there exists a form τ such that ψ ≅ φ ⊥ τ. A hyperbolic plane ℍ is a form isometric to ⟨1, −1⟩. A form φ is called hyperbolic if it is an orthogonal sum of hyperbolic planes. φ is isotropic if it represents 0 nontrivially (i.e., there exists x ∈ V \ {0} such that φ(x) = 0, where V denotes the underlying vector space of φ). Equivalently, φ is isotropic if ℍ ⊂ φ. Otherwise, φ is called anisotropic. By Witt decomposition, any form φ has a unique decomposition (up to isometry) of the shape φ ≅ φ_{an} ⊥ φ_{h} with φ_{an} anisotropic and φ_{h} hyperbolic. The Witt index of φ is then defined to be i_{W} (φ) = \frac{1}{2} dim φ_{h}.

We denote by D_{F} (φ) the set of nonzero elements represented by φ, i.e. D_{F} (φ) = {q(x) | x ∈ V }∩F^{∗}, where V denotes the underlying vector space of φ. For simplicity, we write D_{F} (n) for D_{F} (⟨\underbrace{1, …, 1}_{n}⟩). D_{F} (∞) denotes the set of all nonzero sums of squares in F: D_{F} (∞) = ⋃_{n=1}^{∞} D_{F} (n). By Artin-Schreier theory, for nonreal fields, i.e. fields that do not possess any ordering, one has D_{F} (∞) = F^{∗}. Whereas for real fields F (fields that do have orderings), D_{F} (∞) consists of all elements that are totally positive, i.e. positive with respect to each ordering (see below).

W F will denote the Witt ring of F, and Witt equivalence of two forms φ and ψ (i.e. equality as elements in the Witt ring) will be denoted by φ = ψ. IF is the fundamental ideal of classes of forms of even dimension in W F, and I^{n}F = (IF)^{n} is its n-th power.

A quadratic form of type ⟨1, −a_{1}⟩ ⊗ … ⊗ ⟨1, −a_{n}⟩ (a_{i} ∈ F^{∗}) is called an n-fold Pfister form, and we write ⟨⟨a_{1}, …, a_{n}⟩⟩ for short. P_{n}F (resp. GP_{n}F) denotes the set of all isometry classes of n-fold Pfister forms (resp. of forms similar to n-fold Pfister forms). A form φ is a Pfister neighbor if there exists a Pfister form π and a ∈ F^{∗} such that φ ⊂ aπ and dim φ > \frac{1}{2} dim π (in which case φ is called a Pfister neighbor of π). An important property of Pfister forms is that they are either hyperbolic or anisotropic. Furthermore, if φ is a Pfister neighbor of Pfister forms π and ρ, then π ≅ ρ, and φ is anisotropic iff π is anisotropic.

I^{n}F is additively generated by n-fold Pfister forms. The Arason-Pfister Hauptsatz (APH for short) states that if φ is an anisotropic form in I^{n}F, then dim φ ≥ 2^{n}, and if dim φ = 2^{n}, then in fact φ ∈ GP_{n}F ([AP, Hauptsatz, Kor. 3], [L3, Ch. X, Hauptsatz 5.1, Th. 5.6]).

Let F be a real field and let X_{F} denote its space of orderings. X_{F} is a compact totally disconnected Hausdorff space with a subbasis of the topology given by the clopen sets H(a) = {P ∈ X_{F} | a >_{P} 0}, a ∈ F^{∗}. F is said to satisfy the strong approximation property SAP if given any disjoint closed subsets U, V of X_{F} there exists a ∈ F^{∗} such that U ⊂ H(a) and V ⊂ H(−a). Since nonreal fields have no orderings, such fields are SAP by default.

If φ is a form over F, say, φ ≅ ⟨a_{1}, …, a_{n}⟩, then the signature of φ at an ordering P ∈ X_{F} is defined by

sgn_{P} (φ) = #{i | a_{i} >_{p} 0} − #{i | a_{i} <_{p} 0}

(note that sgn_{P} (φ) as defined above is independent of the chosen diagonalization by Sylvester’s Law of Inertia). By Pfister’s Local-Global Principle, φ being torsion is equivalent to sgn_{P} (φ) = 0 for all P ∈ X_{F}. Note also that any torsion elements in W F are always 2-primary torsion (see, e.g., [L3, Theorem 3.2]). The torsion part of W F is denoted by W_{t}F, and we define I_{t}^{n}F = I^{n}F ∩ W_{t}F. For nonreal fields, we have W F = W_{t}F.

φ is called positive (resp. negative) definite at P ∈ X_{F} if sgn_{P} (φ) = dim φ (resp. sgn_{P} (φ) = − dim φ), and it is called indefinite at P if it is not definite at P. A totally positive definite (t.p.d.) form is a form that is totally definite at each P ∈ X_{F}, similarly one defines totally negative definite (t.n.d.) and totally indefinite (t.i.).

A form φ is called universal if D_{F} (φ) = F^{∗}. Isotropic forms are always universal. If F is real, then x ∈ D_{F} (φ) clearly implies that x >_{P} 0 (resp. x <_{P} 0) if φ is positive (resp. negative) definite at P. If the converse also holds, i.e. if

D_{F} (φ) = {x ∈ F^{∗} | x >_{P} 0 (resp. x <_{P} 0) if φ is

positive (resp. negative) definite at P } .

then φ is called signature-universal (sgn-universal for short). Over a real field we thus have that a form is universal if and only if it is t.i. and sgn-universal. One readily sees that if ũ(F) < ∞ then any form φ with dim φ ≥ ũ(F) is sgn-universal.

A form φ over a real field F is said to have effective diagonalization ED if it has a diagonalization ⟨a_{1}, …, a_{n}⟩ such that H(a_{i}) ⊂ H(a_{i+1}) for 1 ≤ i ≤ n − 1. F is said to be ED if each form over F has ED. Since nonreal fields have no orderings, such fields are ED by default.

The property S_{1} for a real field is defined as follows:

(S_{1}) Every binary torsion form represents a totally positive element (i.e. an ele-

ment in D_{F} (∞)). In addition to u(F) and ũ(F), we will also need the Pythagoras number p(F) (resp.

level s(F)) of a field, the smallest n ∈ ℕ such that each D_{F} (n) = D_{F} (∞) (resp. −1 ∈ D_{F} (n)), provided such an n exists. If no such n exists, one puts p(F) = ∞ (resp. s(F) = ∞). In particular, F is nonreal iff s(F) < ∞ by Artin-Schreier, in which case there exists k ∈ ℕ_{0} such that s(F) = 2^{k}, and one has p(F) ∈ {2^{k}, 2^{k} + 1} (see, e.g., [L3, Ch. XI, Th. 2.2, Th. 5.6]). In contrast, it was shown in [H4] that to any n ∈ ℕ there exists a real field F such that p(F) = n.

3. ED equals SAP plus S_{1}

The following theorem is due to Prestel-Ware [PW]. We give a new proof based mainly on the study of binary forms.

Theorem 3.1. F has ED if and only if F has SAP and S_{1}.

To prove this, we use alternative descriptions of the properties involved.

Lemma 3.2. Let F be a real field.

(i) F is SAP if and only if for all a, b ∈ F^{∗} there exists c ∈ F^{∗} such that H(c) = H(a) ∩ H(b) (or equivalently, there exists d ∈ F^{∗} such that H(d) = H(a) ∪ H(b)).

(ii) F is ED if and only if for all a, b ∈ F^{∗}, there exists c, d ∈ F^{∗} such that

⟨a, b⟩ ≅ ⟨c, d⟩ and H(c) = H(a)∩H(b) (or equivalently, H(d) = H(a)∪H(b)). (iii) F has property S_{1} if and only if, for all a ∈ F^{∗}, s ∈ D_{F} (∞), and x ∈

D_{F} (⟨1, as⟩), there exists t ∈ D_{F} (∞) such that tx ∈ D_{F} (⟨1, a⟩).

Proof. (i) This is well known, see, e.g., [L1, Prop. 17.2].

(ii) The ‘only if’ is nothing else but ED for binary forms. As for the converse, we use induction on the dimension n of forms. Forms of dimension ≤ 2 have ED by assumption. So let φ be a form of dimension n ≥ 3. Then we can write φ = ⟨a_{1}, …, a_{n}⟩ and we may assume that ⟨a_{2}, …, a_{n}⟩ is already an ED. Write ⟨a_{1}, a_{2}⟩ ≅ ⟨b_{1}, b_{2}⟩ with H(b_{1}) = H(a_{1}) ∩ H(a_{2}) (so ⟨b_{1}, b_{2}⟩ is an ED of ⟨a_{1}, a_{2}⟩). Then φ ≅ ⟨b_{1}, b_{2}, a_{3}, …, a_{n}⟩. Now let ⟨c_{2}, …, c_{n}⟩ be an ED of ⟨b_{2}, a_{3}, …, a_{n}⟩. Then one readily checks that ⟨b_{1}, c_{2}, …, c_{n}⟩ is an ED of φ.

(iii) ‘if’: Let ⟨u, v⟩ ≅ u⟨1, uv⟩ be torsion. Then uv = −s with s ∈ D_{F} (∞). Put a = −s. Then ⟨1, −1⟩ ≅ ⟨1, as⟩ which is hyperbolic and hence represents u. But then, by assumption, there exists t ∈ D_{F} (∞) such that tu is represented by ⟨1, a⟩ ≅ ⟨1, −s⟩ and hence t is represented by u⟨1, −s⟩ ≅ ⟨u, v⟩.

‘only if’: x ∈ D_{F} (⟨1, sa⟩) implies that there exists y ∈ F^{∗} such that ⟨1, sa⟩ ≅ ⟨x, y⟩. Now the torsion form xa⟨s, −1⟩ represents some u ∈ D_{F} (∞) by S_{1}. Hence ⟨sa, −a⟩ ≅ ⟨xu, −xus⟩ and hence

⟨1, sa, −a⟩ ≅ ⟨1, xu, −xus⟩ ≅ ⟨−a, x, y⟩

Thus, ⟨1, a⟩ = ⟨x, xus, −xu, y⟩ in W F, so x⟨1, us, −u, xy⟩ is isotropic and there exists v ∈ D_{F} (⟨1, us⟩)∩D_{F} (⟨u, −xy⟩). Note that us ∈ D_{F} (∞), so v ∈ D_{F} (∞). Hence, ⟨1, us⟩ ≅ ⟨v, vus⟩ and ⟨−u, xy⟩ ≅ ⟨−v, vuxy⟩, and we get ⟨1, a⟩ ≅ x⟨vus, vuxy⟩ ≅ ⟨xvus, vuy⟩, thus xt ∈ D_{F} (⟨1, a⟩) with t := vus ∈ D_{F} (∞). □

Proof of Theorem 3.1. ‘only if’: Clearly, ED implies SAP. Now let ⟨a, b⟩ be any binary torsion form. Then sgn_{P} (⟨a, b⟩) = 0, so H(a) ∩ H(b) = ∅, and by ED, there exists c ∈ −D_{F} (∞) and d ∈ D_{F} (∞) such that ⟨a, b⟩ ≅ ⟨c, d⟩, in particular, d is a totally positive element represented by ⟨a, b⟩ and we have established S_{1}.

‘if’: Let F be SAP and S_{1}. We will verify the alternative description of ED from Lemma 3.2(ii). Let ⟨a, b⟩ be any binary form. By SAP, there exists d′ ∈ F^{∗} such that H(a) ∪ H(b) = H(d′). Then ⟨a, b, −d′⟩ is t.i., thus the form φ ≅ ⟨a, b, −d′, −d′ab⟩ ≅ −d′⟨⟨ad′, bd′⟩⟩ has total signature zero and is therefore torsion. Hence, there exists some n ∈ F such that for σ_{n} ≅ ⟨⟨−1⟩⟩^{⊗n} ≅ ⟨1, 1⟩^{⊗n}, we have that σ_{n} ⊗ ⟨a, b, −d′, −d′ab⟩ ∈ GP_{n+2}F is hyperbolic. But then its Pfister neighbor σ_{n} ⊗ ⟨a, b⟩ ⊥ ⟨−d′⟩ is isotropic. It follows that there exist u, v ∈ D_{F} (σ_{n}) ⊂ D_{F} (∞) such that d′ ∈ D_{F} (⟨ua, vb⟩), and hence ad′u ∈ D_{F} (⟨1, abuv⟩). Now uv ∈ D_{F} (∞), and by Lemma 3.2(iii), there exists w ∈ D_{F} (∞) such that ad′uw ∈ D_{F} (⟨1, ab⟩), i.e. d := d′uw ∈ D_{F} (⟨a, b⟩). In particular, there exists c ∈ F^{∗} such that ⟨a, b⟩ ≅ ⟨c, d⟩. Since uw ∈ D_{F} (∞), we have H(d) = H(d′) = H(a) ∪ H(b) as required. □

4. Relations between the Hasse number and the u-invariant

In this section, we will only consider real fields since for nonreal field, u(F) = ũ(F), and most of the statements below are trivially true. It is quite possible for a real field F that u(F) is finite but ũ(F) is infinite. Elman-Prestel [EP, Th. 2.5] gave the following necessary and sufficient criterion for the finiteness of ũ(F):

Theorem 4.1. ũ(F) < ∞ if and only if u(F) < ∞ and F has ED.

Furthermore, in the case ũ(F) < ∞, the value of ũ(F) was related to that of u(F). In fact, it was proved that if ũ(F) is finite, then ũ(F) ≤ (h(F)−1)(u(F)+2) (cf. [EP, Prop. 2.7]). Here, h(F) denotes the height of the field F, i.e. the exponent of the torsion subgroup W_{t}F. If finite, h(F) is the smallest 2-power such that 2^{d} ·W_{t}F = 0. If F is not real, then h(F) = 2D_{F} (∞), and if F is real, then h(F) is the smallest 2-power 2^{d} ≥ p(F) (cf. [L3, Ch. XI, Def. 5.4, Th. 5.6]).

Hornix [Hor1, Th. 3.9] showed that in Elman and Prestel’s bound above one can replace the 2-power h(F) by 2^{b} for any integer b ≥ 0 such that there exists a t.p.d. b-fold Pfister form over F which represents all totally positive elements in F (i.e., which is sgn-universal). Note that the smallest 2^{d} such that the d-fold Pfister form ⟨⟨−1, …, −1⟩⟩ represents all totally positive elements is nothing but h(F), so that Hornix’ result implies the one by Elman and Prestel.

The main purpose of this section is to improve these upper bounds for ũ(F) and at the same time give a new and simplified proof that u(F) < ∞ plus ED is equivalent to ũ(F) < ∞. Let us start with a simple and well known lemma.

Lemma 4.2. For any field F, if p(F) > 2^{n} then ũ(F) ≥ u(F) ≥ 2^{n+1}. In particular, p(F) ≤ u(F) ≤ ũ(F).

Proof. If p(F) > 2^{n}, then there exists x ∈ D_{F} (∞) \ D_{F} (2^{n}), so 2^{n} × ⟨1⟩ ⊥ ⟨−x⟩ is anisotropic. But this is a Pfister neighbor of the form ⟨⟨−1, …, −1, x⟩⟩ ∈ P_{n+1}F which is therefore also anisotropic and which is furthermore torsion. Hence u(F) ≥ 2^{n+1}. It follows immediately that p(F) ≤ u(F) ≤ ũ(F). □

Proposition 4.3. Suppose that F has ED and that there exists an n-dimensional t.p.d. sgn-universal form ρ. Then

ũ(F) ≤ \frac{n}{2}(u(F) + 2) .

Proof. We may clearly assume that u(F) (and hence p(F)) is finite. The form p(F) × ⟨1⟩ is t.p.d. and sgn-universal, so we may assume that n ≤ p(F). If n = 1 then F is obviously pythagorean and u(F) = 0. Since F has ED, any t.i. form φ over F contains a binary torsion form β as a subform. But then β is isotropic as u(F) = 0, hence φ is isotropic. It follows that ũ(F) = 0 and the above inequality is clearly satisfied. So we may assume that 2 ≤ n ≤ p(F) = p and we have ũ(F) ≥ u(F) ≥ p ≥ n by Lemma 4.2.

If ũ(F) = u(F) there is nothing to show. So we may assume that ũ(F) > u(F). Then there exist anisotropic t.i. forms of dimension > u(F) ≥ n. Let φ_{0} be any such form, and write dim φ_{0} = m = rn+k+1 with r ≥ 1 and 0 ≤ k ≤ n−1. Since F is ED and thus SAP, we may assume after scaling that 0 ≤ sgn_{P} φ_{0} ≤ dim φ_{0}−2 = rn+k−1 for all orderings P on F.

Let φ_{1} = a_{0}(φ_{0} ⊥ −ρ)_{an}, where a_{0} is chosen such that 0 ≤ sgn_{P} φ_{1} for all orderings P.

We have i_{W} (φ_{0} ⊥ −ρ) ≤ n − 1, for otherwise one could write φ_{0} ≅ ρ ⊥ τ for some form τ. Since φ_{0} is t.i. and since F has ED, this implies that there exists a totally positive x such that −x is represented by τ. But then the form φ_{0} contains the subform ρ ⊥ ⟨−x⟩ which is isotropic as ρ is t.p.d. and sgn-universal, clearly a contradiction. This implies that

dim φ_{1} ≥ dim φ_{0} + n − 2(n − 1) = (r − 1)n + (k + 1) + 2 .

Note also that sgn_{P} (φ_{0} ⊥ −ρ) = sgn_{P} φ_{0} − n for each ordering P. Hence, one obtains

sgn_{P} φ_{1} ≤ max{(r − 1)n + k − 1, n}

for each ordering P. Note that if r ≥ 2, then φ_{1} is again t.i. as 0 ≤ sgn_{P} φ_{1} < dim φ_{1} for all orderings P. We see that if we apply this procedure altogether r − 1 times, we get a form φ_{r−1} which is anisotropic, t.i., and such that

dim φ_{r−1} ≥ n + (k + 1) + 2(r − 1) ,
0 ≤ sgn_{P} φ_{r−1} ≤ max{n + k − 1, n} for all orderings P.

We therefore have

dim φ_{r−1} − sgn_{P} φ_{r−1} ≥ min{2r, k + 2r − 1} .

Since dim φ_{r−1} − sgn_{P} φ_{r−1} is even, this yields dim φ_{r−1} − sgn_{P} φ_{r−1} ≥ 2r for all orderings P. By ED, the anisotropic form φ_{r−1} contains a torsion subform φ_{t} of dimension ≥ 2r. Hence u(F) ≥ 2r and thus u(F) + 2 ≥ 2(r + 1). On the other hand, by assumption m = rn + k + 1 ≤ n(r + 1). These two inequalities together imply m ≤ \frac{n}{2}(u(F) + 2). It follows readily that ũ(F) ≤ \frac{n}{2}(u(F) + 2). □

Proof of Theorem 4.1. The ‘only if’ part is easy and left to the reader. As for the ‘if’ part, we have ∞ > u(F) ≥ p(F) by Lemma 4.2, and if we put ρ = p(F) × ⟨1⟩, then Proposition 4.3 immediately yields ũ(F) ≤ \frac{p(F)}{2}(u(F) + 2) < ∞. □

If we put ρ = p(F) × ⟨1⟩ as in the proof above, and we let m be such that 2^{m} + 1 ≤ p = p(F) ≤ 2^{m+1}, then the bound of Elman and Prestel yields ũ(F) ≤ (2^{m+1} − 1)(u(F) + 2). Thus, in the ‘worst’ case p = 2^{m+1}, our bound ũ(F) ≤ 2^{m}(u(F) + 2) is almost by a factor 2 better as m gets large. In the ‘best’ case p = 2^{m} + 1, our bound ũ(F) ≤ (2^{m−1} + \frac{1}{2})(u(F) + 2) is almost by a factor 4 better as m gets large.

A similar improvement of the general bound found by Hornix is also obtained this way: Just let ρ be the b-fold Pfister form in Hornix’ result and put n = 2^{b}.

In [GV], an invariant m(F) of the field F has been defined to be the smallest integer n ≥ 1 such that there exists an anisotropic universal n-dimensional form over F. If there is no anisotropic universal form over F, then one defines m(F) = ∞. Note that if F is real, universal forms must necessarily be t.i.. Let us introduce for our purposes another related invariant, m̃(F), which is defined to be the smallest integer n ≥ 1 such that there exists an n-dimensional t.p.d. sgn-universal form.

Again, we put m̃(F) = ∞ if there are no t.p.d. sgn-universal forms. If p(F) < ∞, we have that p(F) × ⟨1⟩ is sgn-universal. Hence m̃(F) ≤ p(F). With this new invariant, Proposition 4.3 immediately implies

Corollary 4.4. Suppose that ũ(F) < ∞. Then

ũ(F) ≤ \frac{m̃(F)}{2}(u(F) + 2) .

Next, we give another bound which will lead to further improvements of the bound by Elman and Prestel.

Proposition 4.5. Suppose that u(F) < ∞ and that F has ED (or, equivalently, that ũ(F) < ∞). Let ρ = ⟨1⟩ ⊥ ρ′ be a t.p.d. m-fold Pfister form, m ≥ 1, such that its pure part ρ′ is sgn-universal. Then

ũ(F) ≤ 2^{m−2}(u(F) + 6) .

If m = 2 then ũ(F) ≤ u(F) + 4.

Proof. If m = 1, then dim ρ′ = 1 and the assumptions imply that F is pythagorean, hence ũ = u = 0 and there is nothing to show. So we may assume m ≥ 2. Furthermore, if d is an integer such that 2^{d} ≤ p(F) = p ≤ 2^{d+1} − 1, then we may assume that m ≤ d + 1. For we have that (2^{d+1} − 1) × ⟨1⟩ is the pure part of ⟨⟨−1, …, −1⟩⟩ ∈ P_{d+1}F and it is totally positive definite and sgn-universal. We proceed similarly as before, but this time we put ũ = ũ(F) = r2^{m} + k + 1 with r ≥ 0 and 0 ≤ k ≤ 2^{m} − 1.

If r = 0 then we have ũ ≤ 2^{m}. If 2^{d} + 1 ≤ p ≤ 2^{d+1} − 1 then u ≥ 2^{d+1} ≥ 2^{m} by Lemma 4.2, and thus necessarily u = ũ and there is nothing to show. Suppose that p = 2^{d} so that in particular u ≥ 2^{d}. Our previous bound yields ũ ≤ 2^{d−1}(u + 2). If m = d + 1, then 2^{d−1}(u + 2) < 2^{m−2}(u + 6) and there is nothing to show. If m ≤ d, then we have ũ = k + 1 ≤ 2^{m} ≤ 2^{d} ≤ u and thus ũ = u, again there is nothing to show. So we may assume that r ≥ 1.

Let φ_{0} be an anisotropic t.i. form of dimension ũ. As before, we may this time assume that dim φ_{0} − 2 = r2^{m} + k − 1 ≥ sgn_{P} φ_{0} ≥ 0 for all orderings P.

We claim that i_{W} (φ_{0} ⊥ −ρ) ≤ 2^{m} − 2. Indeed, otherwise φ_{0} would contain a subform ρ̃ of dimension 2^{m}−1 with ρ̃ ⊂ ρ. Now it is well known that all codimension 1 subforms of a Pfister form are similar to its pure part. Hence, φ_{0} would contain a subform similar to ρ′, and since φ_{0} is t.i. and by ED, φ_{0} would contain a subform similar to ρ′ ⊥ ⟨−x⟩ for some totally positive x. By assumption, ρ′ ⊥ ⟨−x⟩ is isotropic, a contradiction.

Thus, we obtain as in the proof of the previous lemma an anisotropic t.i. form φ_{1} such that

dim φ_{1} ≥ (r − 1)2^{m} + k + 1 + 4 ,
0 ≤ sgn_{P} φ_{1} ≤ max{(r − 1)2^{m} + k − 1, 2^{m}} ,

and reiterating this construction r − 1 times, we get an anisotropic t.i. form φ_{r−1} such that

dim φ_{r−1} ≥ 2^{m} + k + 1 + 4(r − 1) ,
0 ≤ sgn_{P} φ_{r−1} ≤ max{2^{m} + k − 1, 2^{m}} for all orderings P.

This yields dim φ_{r−1} − sgn_{P} φ_{r−1} ≥ 4r − 2 for all orderings P, and thus, by ED, the existence of an anisotropic torsion subform φ_{t} of φ_{r−1} with dim φ_{t} ≥ 4r − 2. In particular, u + 6 ≥ 4(r + 1). On the other hand, ũ ≤ 2^{m}(r + 1) and thus ũ ≤ 2^{m−2}(u + 6).

Now if m = 2, we have dim φ_{r−1} ≥ 4r + k + 1 = dim φ_{0} and 0 ≤ sgn_{P} φ_{r−1} ≤ max{4 + k − 1, 4}. In particular, since all the forms φ_{i} are anisotropic and t.i., it follows readily from the construction and the fact that ũ = 4r + k + 1 that dim φ_{0} = dim φ_{1} = … = φ_{r−1} = ũ. Note also that 0 ≤ k ≤ 3, so that by repeating our construction one more time, we obtain an anisotropic t.i. form φ_{r} such that dim φ_{r} = ũ and sgn_{P} φ_{r} ≤ 4 for all orderings P. Thus, φ_{r} contains a torsion subform of dimension ≥ ũ − 4 and therefore ũ ≤ u + 4. □

Let us investigate how we can use this result to give bounds for ũ in terms of p and u which in certain cases will further improve our previous bound ũ ≤ \frac{p}{2}(u + 2) (and thus the bound found by Elman and Prestel).

Suppose first that p = 2^{d}. Then our first bound yields ũ ≤ 2^{d−1}(u + 2). A priori, we do not know whether there exist t.p.d. m-fold Pfister forms, m ≤ d, whose pure parts are sgn-universal. However, we do know by the definition of p that ⟨⟨−1, …, −1⟩⟩ ∈ P_{d+1}F is such a Pfister form of fold d + 1. But then our second bound only yields ũ ≤ 2^{d−1}(u + 6), which is worse.

Suppose now that d is an integer with 2^{d} + 1 ≤ p ≤ 2^{d+1} − 1. Again, the Pfister form ⟨⟨−1, …, −1⟩⟩ ∈ P_{d+1}F is t.p.d. and its pure part is sgn-universal, so we can use our second bound for m = d + 1. For p = 2^{d} + 1, d ≥ 1, we get 2^{d−1}(u + 6) − \frac{p}{2}(u + 2) = 2^{d+1} − \frac{1}{2}u − 1. In this case, the first bound is better when u ≤ 2^{d+2} − 4 (note that we will have u ≥ 2^{d+1}), they are the same for u = 2^{d+2} − 2, and for u ≥ 2^{d+2} the second bound is sharper.

Let us look a little closer at the case p = 2. Here, our first bound yields ũ ≤ u + 2 if ũ is finite. Under the additional assumption that I_{t}^{3}F = 0, we can show a little more.

Proposition 4.6. Suppose that I_{t}^{3}F = 0, and that u(F) < ∞ and F has ED (or, equivalently, that ũ(F) < ∞). If there exists a t.p.d. sgn-universal binary form ρ over F, then u(F) = ũ(F).

Proof. By [ELP, Th. H], I_{t}^{3}F = 0 implies that ũ = ũ(F) is even. By Proposition 4.3, ũ ≤ u + 2. So let us assume that ũ ≠ u, i.e. ũ = u + 2. The proof of Proposition 4.3 then shows that there exists an anisotropic t.i. form φ (which is nothing but the form φ_{r−1} in the proof) with dim φ = ũ and which contains a torsion subform φ_{t}, dim φ_{t} = dim φ − 2 = u. After scaling, we may assume that φ_{t} ⊥ ⟨1⟩ ⊂ φ. Let d = d_{±}φ_{t}. Then φ_{t} ⊥ ⟨1, −d⟩ ∈ I^{2}F, and since sgn_{P} φ_{t} = 0 and sgn_{P} φ_{t} ⊥ ⟨1, −d⟩ ∈ 4ℤ, it follows that φ_{t} ⊥ ⟨1, −d⟩ ∈ I_{t}^{2}F. As dim φ_{t} ⊥ ⟨1, −d⟩ = u + 2, this form must be isotropic. Thus, φ_{t} ⊥ ⟨1⟩ ≅ ψ ⊥ ⟨d⟩. Comparing discriminants and signatures, it follows that ψ ∈ I_{t}^{2}F. So for every x ∈ F^{∗}, one ⟨1, −x⟩ ⊗ ψ ∈ I_{t}^{3}F = 0, thus ψ ≅ xψ which implies that ψ is universal, hence the subform ψ ⊥ ⟨d⟩ of φ is isotropic, a contradiction. □

The following is an immediate consequence.

Corollary 4.7. Suppose that p(F) = 2 and ũ(F) < ∞. If I_{t}^{3}F = 0 then u(F) = ũ(F). In particular, if u(F) ≤ 6 or ũ(F) ≤ 8, then ũ(F) = u(F).

Let us summarize our bounds for ũ in terms of u and p.

Theorem 4.8. Let F be a real field. Suppose that u(F) < ∞ and that F has ED (or, equivalently, that ũ(F) < ∞). Let m ≥ 2 be an integer.

(i) p(F) = 1 if and only if ũ(F) = u(F) = 0.

(ii) If p(F) = 2 then ũ(F) ≤ u(F) + 2. If in addition I_{t}^{3}F = 0 then ũ(F) =

u(F) = 2n for some integer n ≥ 1. (iii) If p(F) = 3 then ũ(F) ≤ u(F) + 4. (iv) If p(F) = 2^{m} then ũ(F) ≤ 2^{m−1}(u(F) + 2).

(v) If p(F) = 2^{m} + 1 then ũ(F) ≤ (2^{m−1} + \frac{1}{2})(u(F) + 2) if u(F) ≤ 2^{m+2} − 2, and ũ(F) ≤ 2^{m−1}(u(F) + 6) if u(F) ≥ 2^{m+2} − 2.

(vi) If 2^{m} + 2 ≤ p(F) ≤ 2^{m+1} − 1, then ũ(F) ≤ 2^{m−1}(u(F) + 6).

It is, however, difficult to say at this point how good our bounds really are. In fact, we know extremely little about fields with u(F) < ũ(F) < ∞. The only values which could be realized so far are fields where u(F) = 2n and ũ(F) = 2n + 2 for any n ≥ 2 (see [L2], [Hor2], [H6]), and fields with u(F) = 8 and ũ(F) = 12, see [H5, Cor. 6.4].

5. Fields with finite u-invariant and infinite Hasse number

In [EP, §5], one finds examples of non-SAP fields F with prescribed u-invariant 2^{n}, n ≥ 1. These examples were obtained using the method of intersection of henselian fields (cf. [P2]). In this section, we will apply Merkurjev’s method of constructing fields with even u-invariant and modify it in a way such that these fields will be real and such that either they will be non-SAP or they will not have the property S_{1}. In particular, for such fields the Hasse number will be infinite by Theorem 4.1, and it also illustrates the independence of the properties SAP and S_{1}.

Let us first recall some well known results and some special cases of Merkurjev’s index reduction theorem which we will use in the sequel. We refer to [M2], [T] for details. See also [L3, Ch. V.3] for basic results on Clifford invariants c(q) ∈_{2} Br(F) for quadratic forms q over F and how to compute them, and [L3, Ch. X] for basic results on function fields F (q) of quadratic forms q over F.

Lemma 5.1. (i) Let Q_{i} = (a_{i}, b_{i}), 1 ≤ i ≤ n, be quaternion algebras over F with associated norm forms ⟨⟨a_{i}, b_{i}⟩⟩ ∈ P_{2}F. Let A = ⨂_{i=1}^{n} Q_{i} (over F). Then there exist r_{i} ∈ F^{∗}, 1 ≤ i ≤ n, and a form q ∈ I^{2}F, dim q = 2n + 2 such that c(q) = [A] ∈ Br_{2} F and q = ∑_{i=1}^{n} x_{i}⟨⟨a_{i}, b_{i}⟩⟩ in W F. (We will call such a form q an Albert form associated to A.) Furthermore, if A is not Brauer equivalent to a product of < n quaternion algebras (in particular if A is a division algebra), then every Albert form associated to A is anisotropic.

(ii) If q is a form over F with either dim q = 2n + 2 and q ∈ I^{2}F, or dim q = 2n + 1, or dim q = 2n and d_{±}q ≠ 1, then there exist quaternion algebras Q_{i} = (a_{i}, b_{i}), 1 ≤ i ≤ n, such that for A = ⨂_{i=1}^{n} Q_{i} we have c(q) = [A], and there exists an Albert form φ associated to A such that q ⊂ φ.

(iii) If A is a division algebra and if ψ is a form over F of one of the following types:

(a) dim ψ ≥ 2n + 3,

(b) dim ψ = 2n + 2 and d_{±}ψ ≠ 1,

(c) dim ψ = 2n + 2, d_{±}ψ = 1 and c(ψ) ≠ [A] ∈ Br_{2} F,

(d) ψ ∈ I^{3}F, then A stays a division algebra over F (ψ).

Let us also recall some basic facts on the property SAP and weakly isotropic forms which we will use and which are essentially well known. Recall that a form q over F is called weakly isotropic if n × q is isotropic for some n ≥ 1 (over nonreal F, all forms are clearly weakly isotropic as W F = W_{t}F),

Lemma 5.2. (i) F is SAP if and only if for every a, b ∈ F^{∗} the form ⟨1, a, b, −ab⟩

is weakly isotropic. (ii) Suppose that a, b ∈ F^{∗} are such that ⟨1, a, b, −ab⟩ is not weakly isotropic. Let

t ∈ D_{F} (∞). Then ⟨1, a, b, −ab⟩_{F (\sqrt{t})} is not weakly isotropic.

Proof. (i) See [P1, Satz 3.1], [ELP, Th. C].

(ii) Suppose ⟨1, a, b, −ab⟩_{F (\sqrt{t})} is weakly isotropic. Then there exists an integer n ≥ 1 such that n × ⟨1, a, b, −ab⟩_{F (\sqrt{t})} is isotropic. The isotropy over F (\sqrt{t}) implies that n × ⟨1, a, b, −ab⟩ contains a subform similar to ⟨1, −t⟩ (see, e.g., [L3, Ch. VII, Th. 3.1]). Since t is totally positive, it can be written as a sum of, say, m squares in F. But then m × ⟨1, −t⟩ is isotropic. Hence mn × ⟨1, a, b, −ab⟩ is isotropic and thus ⟨1, a, b, −ab⟩ is weakly isotropic. □

Corollary 5.3. Suppose that a, b ∈ F^{∗} are such that ⟨1, a, b, −ab⟩ is not weakly isotropic.

(i) Let F_{pyth} be the pythagorean closure of F (inside some algebraic closure of F). Then ⟨1, a, b, −ab⟩_{F_{pyth}} is not weakly isotropic. In particular, if F is not SAP, then F_{pyth} is not SAP.

(ii) Let ψ be a form over F such that ψ is isotropic over F_{pyth}. Then ⟨1, a, b, −ab⟩_{F (ψ)} is not weakly isotropic. In particular, if F is not SAP, then F (ψ) is not SAP. This is always the case if ψ contains a subform τ, dim τ ≥ 2, such that | sgn_{P} (τ)| ≤ 1 for all orderings P of F.

Proof. (i) follows immediately from the previous lemma and the fact that F_{pyth} can be obtained as the compositum of all extensions K/F (inside an algebraic closure of F) which are of the form F = F_{0} ⊂ F_{1} ⊂ F_{2} ⊂ … ⊂ F_{n} = K for some n, where F_{i+1} = F_{i}(\sqrt{1 + a_{i}^{2}}) for some a_{i} ∈ F_{i}.

(ii) Since ψ is isotropic over F_{pyth}, the extension F_{pyth}(ψ)/F_{pyth} is purely transcendental. Then ⟨1, a, b, −ab⟩_{F_{pyth}(ψ)} is not weakly isotropic because ⟨1, a, b, −ab⟩_{F_{pyth}} is not weakly isotropic and because anisotropic forms (here, n × ⟨1, a, b, −ab⟩_{F_{pyth}}) stay anisotropic over purely transcendental extensions.

Now suppose ψ has a subform τ with dim τ ≥ 2 and | sgn_{P} (τ)| ≤ 1 for all orderings P of F. Since dim τ ≡ sgn_{P} (τ) (mod 2), we have two cases. If sgn_{P} (τ) = 0 for all P, then τ ∈ W_{t}F. Hence τ_{F_{pyth}} is hyperbolic and ψ_{F_{pyth}} is isotropic.

If | sgn_{P} (τ)| = 1 for all P (which implies that dim τ is odd and ≥ 3), then let d ∈ F^{∗} such that q = τ ⊥ ⟨d⟩ ∈ I^{2}F. It follows readily that in fact q = τ ⊥ ⟨d⟩ ∈ I_{t}^{2}F. Thus, q_{F_{pyth}} is hyperbolic and the codimension 1 subform τ_{F_{pyth}} is isotropic. Again, ψ_{F_{pyth}} is isotropic. □

Theorem 5.4. Let N′ be the set of pairs of integers (p, u) such that either p = 1 and u = 0 or u = 2n ≥ 2^{m} ≥ p ≥ 2 for some integers m and n. Let N = N′ ∪ {(p, ∞); p ≥ 2}.

(i) If F is a real field, then (p(F), u(F)) ∈ N. If in addition I_{t}^{k}F = 0 then p(F) ≤ 2^{k−1}.

(ii) Let E be a real field and let (p, u) ∈ N. Then there exists a real field extension F/E such that F is non-SAP, F has property S_{1} and (p(F), u(F)) = (p, u). In particular, ũ(F) = ∞.

(iii) Let E be a real field and let (p, u) ∈ N such that p ≤ 2^{k−1}, k ≥ 1. Then there exists a real field extension F/E such that F is non-SAP, F has property S_{1}, I_{t}^{k}F = 0 and (p(F), u(F)) = (p, u). In particular, ũ(F) = ∞.

Proof. (i) Clearly, u(F) is either even or infinite. It is also obvious that p(F) = 1 implies u(F) = 0. The fact that p(F) ≥ 2 implies the existence of an integer m such that u(F) ≥ 2^{m} ≥ p(F) has been mentioned in the proof of Proposition 4.3. If p(F) > 2^{n−1} then there exists an x ∈ D_{F} (∞) such that 2^{n−1} × ⟨1⟩ ⊥ ⟨−x⟩ is anisotropic. This form is t.i. and a Pfister neighbor of ⟨⟨−1, …, −1, x⟩⟩ ∈ P_{n}F which is therefore torsion and anisotropic. Hence I_{t}^{n}F ≠ 0. This yields the claim.

(ii) First, let us remark that if u(F) ≤ 2, then F automatically has property S_{1}. In fact, S_{1} means that to each torsion binary form β over F there exists an integer n ≥ 1 such that (n × ⟨1⟩) ⊥ β is isotropic. But if u(F) ≤ 2, then ⟨1⟩ ⊥ β is isotropic as it is a Pfister neighbor of some torsion 2-fold Pfister form which itself is hyperbolic as I_{t}^{2}F = 0.

To realize the value (p, u) = (1, 0), let F_{0} be the pythagorean closure of E. Consider the iterated power series field F = F_{0}((x))((y)). By Springer’s theorem (cf. [L3, Ch. VI, §1]), u(F) = 2^{2}u(F_{0}) = 0 and p(F) = p(F_{0}) = 1. Note that we have W_{t}F = I_{t}F = 0. Furthermore, F is not SAP as ⟨1, x, y, −xy⟩ is not weakly isotropic.

To get the non-SAP field F with p(F) = u(F) = 2, let F_{1} = F_{0}(x, y) be the rational function field in two variables. Note that again F_{1} is not SAP as ⟨1, x, y, −xy⟩ is not weakly isotropic. Let φ = ⟨1, −(1 + x^{2})⟩, which is anisotropic and torsion as 1 + x^{2} ∈ D_{F_{1}}(∞) \ F_{1}^{2}. We now construct an infinite tower F_{1} ⊂ F_{2} ⊂ … such that over each F_{i}, φ stays anisotropic and ⟨1, x, y, −xy⟩ will not be weakly isotropic.

The construction is as follows. Having constructed F_{i} with the desired properties, i ≥ 1, let F_{i+1} be the compositum of all function fields of 3-dimensional t.i. forms over F_{i}. Since anisotropic 2-dimensional forms stay anisotropic over the function fields of forms of dimension ≥ 3 (see, e.g. [H1, Th. 1]), φ will stay anisotropic over F_{i+1}. By Cor. 5.3, ⟨1, x, y, −xy⟩ will not be weakly isotropic over F_{i+1}. Now let F = ⋃_{i=1}^{∞} F_{i}. The above shows that φ_{F} is anisotropic so that in particular u(F) ≥ 2, and ⟨1, x, y, −xy⟩_{F} is not weakly isotropic so that F is not SAP. Let q ∈ P_{2}F ∩W_{t}F. Any 3-dimensional subform of q is t.i. and thus isotropic by construction of F. Thus, q is hyperbolic. In particular, I_{t}^{2}F = 0 as I_{t}^{2}F is generated as an ideal by torsion 2-fold Pfister forms (cf. [EL1, Th. 2.8]). By [EL2, Prop. 1.8], this implies u(F) ≤ 2 and thus u(F) = p(F) = 2. Clearly, I_{t}^{2}F = 0.

To get those values (p, u) of N with u ≥ 4, we use a construction quite similar to that in the proofs of [H4, Th. 2, Th. 3].

So let p ≥ 2, F_{1} = F_{0}(x_{1}, x_{2}, …, y_{1}, y_{2}, …) be the rational function field in an infinite number of variables x_{i}, y_{j} over F_{0}. Clearly, F_{1} is not SAP as, for example, the form q = ⟨1, x_{1}, x_{2}, −x_{1}x_{2}⟩ is not weakly isotropic. Let a = 1 + x_{1}^{2} + … + x_{p−1}^{2} and let φ = ⟨\underbrace{1, …, 1}_{p−1}, −a⟩ which is anisotropic by a well known result of Cassels (cf.

[L3, Ch. IX, Cor. 2.4]). Let n ≥ 2 and consider the multiquaternion algebra

A_{n} = (1 + x_{1}^{2}, y_{1}) ⊗ … ⊗ (1 + x_{n−1}^{2}, y_{n−1})

over F_{1}. Then A is a division algebra over F_{1} and it will stay a division algebra over F_{1}(\sqrt{−1}) (see, e.g. [H3, Lem. 2]). By Lemma 5.1, there exists a 2n-dimensional form ψ_{n} such that in W F_{1} we have ψ_{n} = ∑_{i=1}^{n−1} c_{i}⟨⟨1 + x_{i−1}^{2}, y_{i−1}⟩⟩ for suitable c_{i} ∈ F_{1}^{∗}. Since 1+x_{i−1}^{2} ∈ D_{F_{1}}(∞), the forms ⟨⟨1 + x_{i−1}^{2}, y_{i−1}⟩⟩ are torsion and thus ψ_{n} ∈ I_{t}^{2}F_{1}. Furthermore, ψ_{n} is anisotropic as A_{n} is division (this stays true over F_{1}(\sqrt{−1})).

Let now n ≥ 2 and p ≥ 2 be such that 2n ≥ 2^{m} ≥ p for some integer m. Suppose that K is any real field extension of F_{1} such that q_{K} is not weakly isotropic, (A_{n})_{K(\sqrt{−1})} is division and φ_{K} is anisotropic. Consider the following three types of quadratic forms over K:

C_{1}(K) = \{⟨\underbrace{1, …, 1}_{p}, −b⟩ | b ∈ D_{K}(∞)\} ,

C_{2}(K) = \{⟨\underbrace{1, …, 1}_{2p}⟩ ⊥ β | dim β = 2, β ∈ W_{t}F \} ,

C_{3}(K) = {α | α ∈ W_{t}K, dim α ≥ 2n + 2} .

Let ρ ∈ C_{1}(K) ∪ C_{2}(K) ∪ C_{3}(K). Then (A_{n})_{K(ρ)(\sqrt{−1})} is division so that in particular (ψ_{n})_{K(ρ)} is anisotropic. For ρ ∈ C_{i}(K), i = 1, 2, this follows as ρ_{K(\sqrt{−1})} is isotropic (recall that in this case ⟨1, 1⟩ ⊂ ρ) and therefore K(ρ)(\sqrt{−1}) = K(\sqrt{−1})(ρ) is purely transcendental over K(\sqrt{−1}). In the case ρ ∈ C_{3}(K) this is a consequence of Lemma 5.1(iii).

Also, φ_{K(ρ)} is anisotropic. This follows from [H4, Cor.] if ρ ∈ C_{1}(K), and from [H1, Th. 1] by comparing dimensions if ρ ∈ C_{i}(K), i = 2, 3.

q will not be weakly isotropic over K(ρ) by Corollary 5.3.

As before, we now construct a tower of fields F_{1} ⊂ F_{2} ⊂ … as follows. Having constructed F_{i}, we let F_{i+1} be the compositum of all function fields of forms in C_{1}(F_{i}) ∪ C_{2}(F_{i}). Let F = ⋃_{i=1}^{∞} F_{i}. By the above, (ψ_{n})_{F} is anisotropic (and torsion), so that u(F) ≥ 2n. On the other hand, torsion forms of dimension > 2n will be isotropic by construction. Thus u(F) = 2n.

φ_{F} is also anisotropic. Hence p(F) ≥ p. By construction, all forms in C_{1}(F) are isotropic and thus p(F) = p.

q_{F} is not weakly isotropic and therefore F is not SAP. In particular ũ(F) = ∞. Finally, F has property S_{1} as all forms in C_{2}(F) are isotropic by construction. To obtain the values (p, ∞) with p ≥ 2, we do the same construction as before,

but this time only with forms in C_{i}(F), i = 1, 2. This will again yield a non-SAP field F with property S_{1} and with p(F) = p. However, this time we have that (A_{n})_{F} will be a division algebra for each n ≥ 2, so that (ψ_{n})_{F} will be an anisotropic torsion form of dimension 2n for each n ≥ 2. In particular, u(F) = ∞.

(iii) If k ≤ 2 then I_{t}^{2}F = 0 and thus u(F) ≤ 2. These cases have already been dealt with in the proof of (ii). So suppose that k ≥ 3. We repeat the steps in (ii), but when taking composites of function fields, we now include also function fields of forms in

C_{4}(K) = {α | α ∈ I_{t}^{k}K, dim α ≥ 2^{k}}

in addition to those in C_{i}, 1 ≤ i ≤ 3 (resp. C_{1}, C_{2} in the case u = ∞). Since by APH we have that anisotropic forms in I^{k}F must be of dimension ≥ 2^{k}, we immediately see that by construction I_{t}^{k}F = 0.

(A_{n})_{F} will still be a division algebra by Lemma 5.1(iii) as we only consider in addition function fields of forms in I_{t}^{k} with k ≥ 3. Thus, ψ_{n} will be anisotropic as above and we get again that u(F) = u. Since dim φ = p ≤ 2^{k−1}, it follows from [H1, Th. 1] that φ_{F} will still be anisotropic as we only consider in addition function fields of forms which have dimension ≥ 2^{k}. We conclude similarly as above that p(F) = p.

Using the same reasoning as above, Cororollary 5.3 implies that q_{F} is not weakly isotropic and therefore F is not SAP, so that in particular ũ(F) = ∞. Obviously, F will again have the property S_{1}. □

Remark 5.5. In [EP, § 5], examples of real fields F with u(F) = 2^{n} have been constructed for each integer n ≥ 1 with the property that u(F (\sqrt{a})) = ∞ and p(F (\sqrt{a})) = 2. u(F (\sqrt{a})) = ∞ implies that F is non-SAP by [EP, Cor. 2.4]. It is also indicated how to obtain such a field which does not satisfy S_{1} (resp. certain properties S_{n} which generalize S_{1}), see [EP, Rem. 5.3].

We will now construct real SAP fields F such that ũ(F) = ∞ and u(F) = 2n for a given n. First, we note that it will be impossible to realize such examples for all values in N (cf. Theorem 5.4).

Proposition 5.6. Let F be real and SAP. If u(F) ≤ 2 then u(F) = ũ(F).

Proof. As remarked in the proof of Theorem 5.4, u(F) ≤ 2 implies that F has property S_{1}. Since F is SAP by assumption, we thus have ũ(F) < ∞. Now p(F) ≤ u(F) ≤ 2, and by Theorem 4.8 we have u(F) = ũ(F). □

Theorem 5.7. Let N be as in Theorem 5.4.

(i) If F is a real SAP field with ũ(F) = ∞, then u(F) ≥ 4 and (p(F), u(F)) ∈ N. Furthermore, I_{t}^{2}F ≠ 0. If in addition I_{t}^{k}F = 0, k ≥ 3, then p(F) ≤ 2^{k−1}.

(ii) Let E be a real field and let (p, u) ∈ N with u ≥ 4. Then there exists a real field extension F/E such that F is SAP, F does not have property S_{1} and (p(F), u(F)) = (p, u). In particular, ũ(F) = ∞.

(iii) Let E be a real field and let (p, u) ∈ N with u ≥ 4 and such that p ≤ 2^{k−1}, k ≥ 3. Then there exists a real field extension F/E such that F is SAP, F does not have property S_{1}, I_{t}^{k}F = 0 and (p(F), u(F)) = (p, u). In particular, ũ(F) = ∞.

Proof. (i) If I_{t}^{2}F = 0, then u(F) ≤ 2 by [EL2, Prop. 1.8]. The result now follows from Theorem 5.4 and Proposition 5.6.

(ii) We proceed as in the proof of Theorem 5.4(ii) for the case (p, u) ∈ N and 2n = u ≥ 4, except for the definition of F_{1}, which now will be the power series field in one variable t over the field which was denoted by F_{1} in the proof of Theorem 5.4(ii): F_{1} = F_{0}(x_{1}, x_{2}, …, y_{1}, y_{2}, …)((t)). We keep the notations for A_{n}, ψ_{n}, C_{1}(K), C_{3}(K). We redefine C_{2}(K):

C_{2}(K) = {⟨1, 1⟩ ⊗ ⟨1, x, y, −xy⟩ | x, y ∈ K^{∗}} .

We construct a tower of fields F_{1} ⊂ F_{2} ⊂ … as follows. Having constructed F_{i}, we let F_{i+1} be the compositum of all function fields of forms in C_{1}(F_{i}) ∪ C_{2}(F_{i}) ∪ C_{3}(F_{i}). Let F = ⋃_{i=1}^{∞} F_{i}.

Exactly as in the proof of Theorem 5.4(ii), it follows that (u(F), p(F)) = (p, u). It remains to show that F is SAP and does not have property S_{1}.

Now by construction, for all x, y ∈ F^{∗} we have that ⟨1, 1⟩ ⊗ ⟨1, x, y, −xy⟩ is isotropic. In particular, each form ⟨1, x, y, −xy⟩ is weakly isotropic, which shows by Lemma 5.2 that F is SAP.

Now let d = 1 + x_{1}^{2} and consider the form μ_{m} = m × ⟨1⟩ ⊥ t⟨1, −d⟩ which is anisotropic over F_{1} by Springer’s theorem. Let L_{1} = F_{1} and L′_{1} = F′_{1} = F_{0}(x_{1}, x_{2}, …, y_{1}, y_{2}, …). We now construct a tower of fields L_{1} ⊂ L_{2} ⊂ … such that L_{i} will be the power series field in the variable t over some L′_{i}, L_{i} = L′_{i}((t)), such that F_{i} ⊂ L_{i}, and (μ_{m})_{L_{i}} anisotropic for all m ≥ 0 and all i ≥ 1. This then shows that (μ_{m})_{F_{i}} is anisotropic for all m ≥ 0, i ≥ 1, and therefore (μ_{m})_{F} will be anisotropic for all m ≥ 0. It follows that the torsion form (−t⟨1, −d⟩)_{F} does not represent any element in D_{F} (∞). Thus, F does not have property S_{1}.

Suppose we have constructed L_{i} = L′_{i}((t)). Note that necessarily L_{i} is real as (μ_{m})_{L_{i}} is anisotropic for all m ≥ 0. Let P_{i} ∈ X_{L′_{i}} be any ordering and M′_{i} be the compositum over L′_{i} of the function fields of all forms (defined over L′_{i}) in

C′(L′_{i}) = {α | α indefinite at P_{i}, dim α ≥ 3} .

Let M_{i} = M′_{i}((t)).

Now let ρ ∈ C_{1}(F_{i}) ∪ C_{2}(F_{i}) ∪ C_{3}(F_{i}) and consider L_{i}(α). By Springer’s theorem, ρ_{L_{i}} ≅ β ⊥ tγ where β, γ are defined over L′_{i}. Suppose ρ ∈ C_{1}(F_{i}). Then ρ ≅ p×⟨1⟩ ⊥ ⟨−b⟩ with b ∈ D_{L_{i}}(∞). But then, up to a square, b ∈ D_{L′_{i}}(∞) and thus ρ_{L_{i}} ∈ C′(L′_{i}). Hence, ρ_{M_{i}} is isotropic and therefore M_{i}(ρ)/M_{i} is purely transcendental.

Suppose ρ = ⟨1, 1⟩ ⊗ ⟨1, x, y, −xy⟩ ∈ C_{2}(F_{i}). Then either ρ_{L_{i}} is already defined over L′_{i}, in which case it is a t.i. form of dimension 8 and thus in C′(L′_{i}). Or there exist a, b ∈ L_{i}^{′∗} such that ρ ≅ ⟨1, 1⟩ ⊗ ⟨1, a⟩ ⊥ bt⟨1, 1⟩ ⊗ ⟨1, −a⟩. then either ⟨1, 1⟩ ⊗ ⟨1, a⟩ is indefinite at P_{i} and thus in C′(L′_{i}), or ⟨1, 1⟩ ⊗ ⟨1, −a⟩ is indefinite at P_{i} and thus in C′(L′_{i}). In any case, we see that ρ_{M_{i}} is isotropic, and again M_{i}(ρ)/M_{i} is purely transcendental.

Finally, suppose that ρ ∈ C_{3}(F_{i}). Then ρ_{L_{i}} ∈ W_{t}L_{i}, and if we write ρ ≅ β ⊥ tγ with β and γ defined over L′_{i}, then β ∈ W_{t}L′_{i} and γ ∈ W_{t}L′_{i}. Now dim ρ ≥ 6, and hence dim β ≥ 4 or dim γ ≥ 4. Hence β ∈ C′(L′_{i}) or γ ∈ C′(L′_{i}). As above, we conclude that ρ_{M_{i}} is isotropic and that M_{i}(ρ)/M_{i} is purely transcendental.

Now let N_{i} be the compositum of the function fields of all forms α_{M_{i}} with α ∈ C_{1}(F_{i}) ∪ C_{2}(F_{i}) ∪ C_{3}(F_{i}). By the above, N_{i}/M_{i} is purely transcendental. Let B be a transcendence basis so that N_{i} = M_{i}(B) = M′_{i}((t))(B). We now put L′_{i+1} = M′_{i}(B) and L_{i+1} = L′_{i+1}((t)) = M′_{i}(B)((t)). There are obvious inclusions F_{i+1} ⊂ N_{i} = M′_{i}((t))(B) ⊂ M′_{i}(B)((t)) = L_{i+1}. Since M′_{i} is obtained from L′_{i} by taking function fields of forms indefinite at P_{i}, we see that P_{i} extends to an ordering on M′_{i} and thus clearly also to orderings on L_{i+1}.

It remains to show that μ_{m} stays anisotropic over L_{i+1}. Now m × ⟨1⟩ is clearly anisotropic over the real field L′_{i+1}. Also, ⟨1, −d⟩, which is anisotropic over L′_{i} by assumption, stays anisotropic over L′_{i+1} as L′_{i+1} is obtained by taking function fields of forms of of dimension ≥ 3 over L′_{i} followed by a purely transcendental extension. By Springer’s theorem, (μ_{m})_{L_{i+1}} = (m × ⟨1⟩ ⊥ t⟨1, −d⟩)_{L_{i+1}} is anisotropic.

To get the values of type (p, ∞), we adjust the above arguments as in the proof of Theorem 5.4(iii).

(iii) This follows easily by combinig the proof of part (ii) above with that of Theorem 5.4(iii). We leave the details to the reader. □

Remark 5.8. Let K be any real field over E with u(K) = 2n and such that K is uniquely ordered. For n ≥ 2, such fields have been constructed in [H4, Th. 2]. The construction there can also readily be used to get such a K for n = 1.

Now consider F = K((t)), the power series field in one variable t over K. By Springer’s theorem, u(F) = 4n = 2u(K). Since K is uniquely ordered, we have that F is SAP (cf. [ELP, Prop. 1]). Since u(K) > 0, K is not pythagorean. So let d ∈ D_{K}(∞) \ K^{∗2}. Then the form (m × ⟨1⟩) ⊥ t⟨1, −d⟩ is anisotropic for all m (again by Springer’s theorem), and since t⟨1, −d⟩ is torsion, we see that F does not have property S_{1}. Hence ũ(F) = ∞.

This rather simple construction yields SAP fields with u(F) = 4n and ũ(F) = ∞ for all n ≥ 1, but it does not provide examples where u(F) = 4n + 2, n ≥ 1. Furthermore, one checks easily that it will not yield examples of SAP fields with ũ(F) = ∞, u(F) > 4 and I_{t}^{3}F = 0, which do exist by the above theorem.

6. Linkage of fields and the Pfister neighbor property

Let us recall the notion of linkage of fields.

Definition 6.1. Let n ≥ 1 be an integer. A field F is called n-linked if to any n-fold Pfister forms π_{1} and π_{2} over F there exist an a ∈ F^{∗} and (n − 1)-fold Pfister forms σ_{1} and σ_{2} such that π_{i} ≅ ⟨⟨a⟩⟩ ⊗ σ_{i}, i = 1, 2. F is called linked if F is 2-linked.

Obviously, every field is 1-linked, so the property of n-linkage is only interesting for n ≥ 2 The following properties are well known, [EL3, § 2], [H2].

Proposition 6.2. Let n ≥ 2. The following are equivalent.

(i) F is n-linked.

(ii) F is m-linked for all m ≥ n.

(iii) The classes of n-fold Pfister forms in I^{n}F/I^{n+1}F form an additive subgroup

of I^{n}F/I^{n+1}F. (iv) To each form φ ∈ I^{n}F there exists a form π ∈ P_{n}F such that φ ≡ π mod

I^{n+1}F.

(v) Each anisotropic form φ ∈ I^{n}F can be written as φ ≅ π_{1} ⊥ … ⊥ π_{r} with π_{i} ∈ GP_{n}F, 1 ≤ i ≤ r (φ is then said to have simple decomposition).

(vi) To each anisotropic form φ ∈ I^{n}F there exists a form σ ∈ P_{n−1}F and an even-dimensional form τ over F such that φ ≅ σ ⊗ τ.

Furthermore, if F is n-linked, n ≥ 2, then I_{t}^{n+2}F = 0.

We can extend this list of equivalences further for linked (i.e. 2-linked) fields. These equivalent formulations for the linkage property are essentially well known, so we will only sketch the proofs for some of them and give the relevant references for the others.

Proposition 6.3. The following properties of a field F are equivalent :

(i) F is linked.

(ii) If Q_{1} and Q_{2} are quaternion algebras over F then there exist a, b_{1}, b_{2} ∈ F^{∗}

such that Q_{i} ≅ (a, b_{i})_{F}, i = 1, 2, i.e. Q_{1} and Q_{2} have a common slot. (iii) The classes of quaternion algebras over F form a subgroup of the exponent-

2-part _{2}Br(F) of the Brauer group of F. (iv) To every element [A] ∈ _{2}Br(F) there exists a quaternion algebra (a, b)_{F},

a, b ∈ F^{∗}, such that [A] = [(a, b)_{F}].

(v) Every 6-dimensional form in I^{2}F is isotropic.

(vi) Every 5-dimensional form over F is a Pfister neighbor.

(vii) To every anisotropic form φ in I^{2}F there exists a quadratic extension L/F such that φ_{L} is hyperbolic.

Proof. (i)⇔(ii). Let Q_{i} = (u_{i}, v_{i})_{F}, u_{i}, v_{i} ∈ F^{∗}, i = 1, 2, and let π_{i} = ⟨⟨u_{i}, v_{i}⟩⟩ ∈ P_{2}F be the norm form associated to Q_{i}. The equivalence of (i) and (ii) follows readily from the fact that two quaternion algebras are isomorphic if and only if their associated norm forms are isometric, cf. [L3, Ch. III, Th. 2.5], [S, Ch. 2, Th. 11.9].

(ii)⇒(iii) is clear.

(iii)⇒(iv) follows readily from Merkurjev’s theorem [M1] which implies that every element in _{2}Br(F) is Brauer equivalent to a product of quaternion algebras.

(iv)⇒(ii). Let Q_{1} and Q_{2} be quaternion algebras over F. By assumption, there exists a quaternion algebra Q_{3} such that [Q_{1} ⊗ Q_{2} ⊗ Q_{3}] = 1 ∈ _{2}Br(F). A well known theorem of Albert then states that there exist a, b_{1}, b_{2}, b_{3} ∈ F^{∗} such that Q_{i} ≅ (a, b_{i})_{F}, 1 ≤ i ≤ 3 (cf. [S, Ch. 2, Th. 13.6]).

(i)⇔(v). Up to a scalar multiple, a 6-dimensional I^{2}-form φ can be written as φ ≅ ⟨−x, −y, xy, u, v, −uv⟩, x, y, u, v ∈ F^{∗}. The equivalence now follows readily from the fact that φ is isotropic iff ⟨−x, −y, xy⟩ and ⟨−u, −v, uv⟩ represent a common element iff there exist a, b_{1}, b_{2} ∈ F^{∗} such that ⟨⟨x, y⟩⟩ ≅ ⟨⟨a, b_{1}⟩⟩ and ⟨⟨u, v⟩⟩ ≅ ⟨⟨a, b_{2}⟩⟩.

(v)⇔(vi) Let φ be a 5-dimensional form over F and let d ∈ F^{∗}. Then φ ⊥ ⟨−d⟩ ∈ I^{2}F iff d = det(φ) ∈ F^{∗}/F^{∗2}. The equivalence is now an easy consequence of the well known fact that φ is a Pfister neighbor iff φ represents d iff φ ⊥ ⟨−d⟩ is isotropic (see, e.g., [L3, Ch. X, Prop. 4.19]).

(i)⇔(vii). Let L = F (\sqrt{a}) be a quadratic extension, and let φ be an anisotropic form over F. Then it is well known that φ_{L} is hyperbolic iff there exists a form τ over F such that φ ≅ ⟨⟨a⟩⟩ ⊗ τ (see, e.g., [L3, Ch. VII, Th. 3.2], [S, Ch. 2, Th. 5.2]).

The equivalence corresponds therefore to the equivalence (i)⇔(vi) in the previous proposition for n = 2. □

Linked fields have a very well understood theory of quadratic forms. In particular, we have a complete knowledge of the possible values of the u-invariant and the Hasse number of linked fields. We will now state these results which are due to Elman and Lam [EL3] and Elman [E, Th. 4.7].

Theorem 6.4. Let F be a linked field. Then u(F) = ũ(F) ∈ {0, 1, 2, 4, 8}. In particular, I_{t}^{4}F = 0. Furthermore, let n ∈ {0, 1, 2}. Then ũ(F) ≤ 2^{n} iff I_{t}^{n+1}F = 0.

Note that u(F) = ũ(F) = 0 can only occur when F is real, whereas u(F) = ũ(F) = 1 implies that F is nonreal.

It is a natural question to ask whether one can obtain also good information (for example, upper bounds) on u and ũ after weakening the linkage assumption on F. One possible way of generalizing is to assume F to be n-linked for some n ≥ 2. For n = 2 we have linkage and the above result. Suppose F is nonreal and I^{3}F = 0. Then obviously F is 3-linked as there are no anisotropic 3-fold Pfister forms in the first place. However, Merkurjev constructed to each positive even integer n a field F with I^{3}F = 0 and u(F) = n (resp. a field with I^{3}F = 0 and u(F) = ∞), [M2]. This shows that if we weaken the linkage assumption by only assuming n-linkage for some n ≥ 3, the u-invariant can become arbitrarily large and even infinite.

One of the main purposes of this paper is to show that item (vi) in Proposition 6.3 can be generalized suitably to yield meaningful bounds on the Hasse number resp. the u-invariant.

Definition 6.5. Let n ≥ 0 be an integer. A field F is said to have the Pfister neighbor property of order n, P N(n) for short, if every form over F of dimension 2^{n} + 1 is a Pfister neighbor.

Remark 6.6. (i) Every field has property P N(0) and P N(1). F has property

P N(2) iff F is linked, see Proposition 6.3. (ii) Let n ≥ 2. Every isotropic form of dimension 2^{n} + 1 is a Pfister neighbor. In

fact, if dim φ = 2^{n}+1 and φ is isotropic, then φ ≅ ℍ ⊥ ψ with dim ψ = 2^{n}−1. Then φ ⊥ −ψ ≅ π ∈ P_{n+1}F, where π denotes the hyperbolic (n + 1)-fold Pfister form.

Lemma 6.7. Let n ≥ 2.

(i) F has property P N(n) if and only if there exists to every form φ over F a form ψ such that dim ψ ≤ 2^{n} if dim φ even (resp. dim ψ ≤ 2^{n} − 1 if dim φ odd) such that φ ≡ ψ mod I^{n+1}F.

(ii) If F has property P N(n), n ≥ 2, then F is n-linked. In particular, I_{t}^{n+2}F = 0. Furthermore, F is ED.

Proof. (i) ‘only if’: If dim φ ≤ 2^{n}, then put ψ ≅ φ. So suppose dim φ ≥ 2^{n} + 1. Write φ ≅ ψ ⊥ τ with dim ψ = 2^{n} + 1. By P N(n), ψ is a Pfister neighbor and there exists ψ′, dim ψ′ = 2^{n} − 1 such that ψ ⊥ −ψ′ ≅ π ∈ GP_{n+1}F. Then, in W F, we have

φ ≡ φ − π ≡ ψ′ ⊥ τ mod I^{n+1}F .

Now dim ψ′ ⊥ τ = dim φ − 2 and the result follows by an easy induction on the dimension.

‘if’: Let dim φ = 2^{n} + 1. By assumption, there exists a form ψ, dim ψ ≤ 2^{n} − 1 such that φ ⊥ −ψ ∈ I^{n+1}F. If dim ψ < 2^{n} −1, then by APH, φ ⊥ −ψ is hyperbolic, i.e. φ = ψ ∈ W F, and φ is isotropic by comparing dimensions, hence a Pfister neighbor by Remark 6.6(ii). If dim ψ = 2^{n} − 1 then dim(φ ⊥ −ψ) = 2^{n+1} and thus φ ⊥ −ψ ∈ GP_{n+1}F, again by APH, which implies that φ is a Pfister neighbor.

(ii) To show that F is n-linked, let φ ∈ I^{n}F. By (i), there exists ψ such that dim ψ ≤ 2^{n} and φ ≡ ψ mod I^{n+1}F. But clearly ψ ∈ I^{n}F, and by APH we have that either ψ is hyperbolic (in which case we may replace it by the n-fold hyperbolic Pfister form), or ψ is anisotropic and of dimension 2^{n}, in which case ψ ∈ GP_{n}F. Let x ∈ F^{∗} such that xψ ∈ P_{n}F. We then have ψ ≡ xψ mod I^{n+1}F, and n-linkage together with I_{t}^{n+2}F = 0 follows from Proposition 6.2.

Now n-linked fields, n ≥ 2, are easily seen to be SAP. So to establish ED, it suffices to establish property S_{1} by Theorem 3.1. Let ⟨a, b⟩ be any torsion form. Let γ ≅ ⟨\underbrace{1, …, 1}_{2^{n}−1}⟩. Then by P N(n), the form γ ⊥ ⟨−a, −b⟩ is a t.i. Pfister neighbor

of a Pfister form π ∈ P_{n+1}F. Since π contains γ which is a Pfister neighbor (and in fact subform) of σ_{n} ≅ ⟨1, 1⟩^{⊗n}, one necessarily has that σ_{n} divides π, so there exists c ∈ F^{∗} such that π ≅ σ_{n} ⊗ ⟨1, c⟩. Now π contains a t.i. Pfister neighbor and is therefore also t.i. and hence torsion. But then ρ ≅ ⟨1, 1⟩ ⊗ σ_{n} ⊗ ⟨1, c⟩ ∈ P_{n+2}F is torsion as well and therefore hyperbolic by Proposition 6.2. Now σ_{n} ⊥ γ ⊥ ⟨−a, −b⟩ is a Pfister neighbor of ρ. Since ρ is hyperbolic, its neighbor σ_{n} ⊥ γ ⊥ ⟨−a, −b⟩ is isotropic. Hence there exists x ∈ D_{F} (⟨a, b⟩) ∩ D_{F} (σ_{n} ⊥ γ). But clearly, D_{F} (σ_{n} ⊥ γ) ⊂ D_{F} (∞) which shows that the binary torsion form ⟨a, b⟩ represents the totally positive element x. □

The following observation is essentially due Fitzgerald [F, Lemma 4.5(ii)].

Lemma 6.8. Suppose that ũ(F) ≤ 2^{n}. Let φ be a form over F of dimension 2^{n} + 1. Then φ is a Pfister neighbor. In particular, F has P N(n).

Proof. Since ũ(F) < ∞ implies that F is SAP, we may assume that after scaling, sgn_{P} (φ) ≥ 0 for all P ∈ X_{F}, and that there exists c ∈ F^{∗} such that H(c) = {P ∈ X_{F} | sgn_{P} (φ) = dim φ\}. In particular, the Pfister form ⟨⟨\underbrace{−1, …, −1}_{n}, −c⟩⟩ ∈ P_{n+1}F

is positive definite at all those P ∈ X_{F} at which φ is positive definite, and it has signature zero at all those P ∈ X_{F} at which φ is indefinite. Let ψ ≅ (π ⊥ −φ)_{an}. It follows that | sgn_{P} (ψ)| ≤ 2^{n}−1 for all P ∈ X_{F}. But since ũ(F) ≤ 2^{n}, the anisotropic form ψ must therefore have dim ψ ≤ 2^{n}, so in particular,

i_{W} (π ⊥ −φ) = \frac{1}{2}(dim(π ⊥ −φ) − dim ψ) ≥ \frac{1}{2}(2^{n+1} + 1) ,

and therefore i_{W} (π ⊥ −φ) ≥ 2^{n} + 1 = dim φ, which implies that φ ⊂ π. In particular, φ is a Pfister neighbor of π. □

Theorem 6.9. If a field F has property P N(n), n ≥ 2, then either u(F) ≤ ũ(F) ≤ 2^{n}, or 2^{n+1} ≤ u(F) ≤ ũ(F) ≤ 2^{n+1} + 2^{n} − 2.

Proof. Let F be a field with property P N(n) for some n ≥ 2. Suppose that ũ(F) > 2^{n}, i.e. there exists an anisotropic t.i. φ with dim φ = m > 2^{n}. By Lemma 6.7, F has ED and so φ can be diagonalized as φ ≅ ⟨a_{1}, …, a_{m}⟩ with −a_{1}, a_{m} ∈ D_{F} (∞). By removing some of the a_{i}, 2 ≤ i ≤ m − 1 if necessary, we will retain a t.i. form, so we may assume that φ is t.i. and dim φ = 2^{n} + 1. But then, by P N(n), φ is a Pfister neighbor of some π ∈ P_{n+1}F which in turn is torsion and anisotropic as its Pfister neighbor φ is t.i. and anisotropic. This shows that 2^{n+1} ≤ u(F) ≤ ũ(F).

Now suppose that ũ(F) > 2^{n+1} + 2^{n} − 2. By a similar argument as above, we conclude that there exists an anisotropic t.i. form φ with dim φ = 2^{n+1} + 2^{n} − 1. By Lemma 6.7, there exists an anisotropic form ψ of dimension ≤ 2^{n} − 1 such that φ ≡ ψ mod I^{n+1}F. Let π ≅ (φ ⊥ −ψ)_{an} ∈ I^{n+1}F. Then by dimension count and since φ is anisotropic, we have 2^{n+1} ≤ dim π ≤ 2^{n+2} − 2. But since F is (n + 1)-linked by Lemma 6.7(ii) and Proposition 6.2, anisotropic forms in I^{n+1}F have dimension divisible by 2^{n+1}, so we have dim π = 2^{n+1}, which, by APH, implies that π ∈ GP_{n+1}F. Also, φ = π ⊥ ψ in W F, and by dimension count we have in fact φ ≅ π ⊥ ψ.

After scaling, we may assume that π ∈ P_{n+1}F, so that sgn_{P} (π) ∈ {0, 2^{n+1}}. Now φ is t.i., and since F has ED by Lemma 6.7(ii), we can write ψ ≅ ⟨a, …⟩ with a <_{P} 0 whenever sgn_{P} (π) = 2^{n+1}. But then π ⊥ ⟨a⟩ is a t.i. subform of φ. On the other hand, π ⊥ ⟨a⟩ is also a Pfister neighbor of π ⊗⟨1, a⟩ ∈ P_{n+2}F. Since π ⊥ ⟨a⟩ is t.i., this implies that π ⊗ ⟨1, a⟩ is torsion and therefore hyperbolic since I_{t}^{n+2}F = 0 by Proposition 6.2 and Lemma 6.7(ii). But then the Pfister neighbor π ⊥ ⟨a⟩ is isotropic and therefore also φ, a contradiction. □

We now obtain the equivalence of (i) and (iii) in Theorem 1.1

Corollary 6.10. ũ(F) < ∞ if and only if F has P N(n) for some n ≥ 2.

Proof. The ‘if’-part follows from Theorem 6.9, the converse from Lemma 6.8. □

Remark 6.11. If F is real, then we still get a sufficient criterion for the finiteness of u(F) even if ũ(F) = ∞. Indeed, for real F, one has that if u(F (\sqrt{−1})) is finite then u(F) is finite, more precisely, one has u(F) < 4u(F (\sqrt{−1})) (see [EKM, Th. 37.4]). Thus, we get the following: If F (\sqrt{−1}) has property P N(n) for some n ≥ 2, then u(F) < 2^{n+3} + 2^{n+2} − 8.

Conjecture 6.12. If a field F has property P N(n), n ≥ 2, then u(F) ≤ ũ(F) ≤ 2^{n}, or u(F) = ũ(F) = 2^{n+1}.

Corollary 6.13. For n ≥ 2, P N(n) implies P N(m) for all m ≥ n+2. Furthermore, the following are equivalent:

(i) Conjecture 6.12 holds.

(ii) For n ≥ 2, P N(n) implies P N(n + 1).

Proof. If n ≥ 2, then P N(n) implies that ũ(F) ≤ 2^{n+2}, and P N(m) for m ≥ n + 2 follows from Lemma 6.8.

Now suppose that F has P N(n) and that Conjecture 6.12 holds. Then P N(n+1) follows from Lemma 6.8. Conversely, suppose that n ≥ 2 and that P N(n) implies P N(n+1). Then we have u(F) ≤ ũ(F) ≤ 2^{n} or 2^{n+1} ≤ u(F) ≤ ũ(F) ≤ 2^{n+1}+2^{n}−2 because of P N(n), and also u(F) ≤ ũ(F) ≤ 2^{n+1} or 2^{n+2} ≤ u(F) ≤ ũ(F) ≤ 2^{n+2} + 2^{n+1} − 2 because of P N(n + 1). Putting the two together, we obtain u(F) ≤ ũ(F) ≤ 2^{n} or u(F) = ũ(F) = 2^{n+1}. □

The only evidence we have as to the veracity of Conjecture 6.12 is the following.

Lemma 6.14. P N(2) implies P N(3). In particular, if F has P N(2), then u(F) ≤ ũ(F) ≤ 4 or u(F) = ũ(F) = 8.

Proof. Suppose F has P N(2) and let φ be any 9-dimensional form over F. Write φ ≅ α ⊥ β with dim α = 5. Since α is a Pfister neighbor, there exists π ∈ GP_{2}F such that π ⊂ α ⊂ φ (see, e.g., [L3, Ch. X, Prop. 4.19]). Write φ ≅ π ⊥ γ. Then dim γ = 5 and γ is also a Pfister neighbor, so there exists ρ ∈ GP_{2}F such that ρ ⊂ γ. Hence, there exist a, b, c, d, e, f, g ∈ F^{∗} such that φ ≅ a⟨⟨b, c⟩⟩ ⊥ d⟨⟨e, f⟩⟩ ⊥ ⟨g⟩.

Since P N(2) implies that F is linked by Proposition 6.3, we may assume that b = e, and after scaling (which doesn’t change the property of being a Pfister neighbor), we may also assume a = 1, so

φ ≅ ⟨⟨b, c⟩⟩ ⊥ d⟨⟨b, f⟩⟩ ⊥ ⟨g⟩ ⊂ ⟨⟨b⟩⟩ ⊗ (⟨⟨c⟩⟩ ⊥ d⟨⟨f⟩⟩ ⊥ ⟨g⟩) .

Now δ ≅ ⟨⟨c⟩⟩ ⊥ d⟨⟨f⟩⟩ ⊥ ⟨g⟩ has dimension 5 and is therefore again a Pfister neighbor, so as above there exist h, k, l, m ∈ F^{∗} such that δ ≅ h⟨⟨k, l⟩⟩ ⊥ ⟨m⟩. We thus get that

φ ⊂ ⟨⟨b⟩⟩ ⊗ δ ≅ h⟨⟨b, k, l⟩⟩ ⊥ m⟨⟨b⟩⟩ ⊂ h⟨⟨b, k, l, −hm⟩⟩ ∈ GP_{4}F ,

which shows that φ is a Pfister neighbor.

The remaining statement now follows from Corollary 6.13. □

Using this, we are now able to give our proof of the Elman-Lam result.

Proof of Theorem 6.4. Let F be linked. By Proposition 6.3, this is equivalent to F having P N(2). By Lemma 6.14, we have that u(F) ≤ ũ(F) ≤ 4 or u(F) = ũ(F) = 8.

All that remains to be shown is that ũ(F) ≤ 4 implies ũ(F) ≠ 3 and u(F) = ũ(F).

Now over any ED field, any t.i. form of dimension n ≥ 3 contains a t.i. subform of dimension n − 1. Thus, we readily conclude that if 3 ≤ ũ(F) ≤ 4 then there exists an anisotropic t.i. form φ of dimension 3. But then φ is a Pfister neighbor of some anisotropic π ∈ P_{2}F that is t.i. and thus torsion, so we readily get ũ(F) ≥ u(F) ≥ 4 and thus ũ(F) = u(F) = 4. Finally, it is clear that if ũ(F) ≤ 2 then ũ(F) = u(F). □

Example 6.15. (i) In [B1], Becher studies fields F that possess an anisotropic form φ such that any other anisotropic form over F is a subform of φ. It can be shown that such a form φ is then necessarily an n-fold Pfister form for some n ∈ ℕ_{0} (called supreme Pfister form), in which case F is nonreal and u(F) = dim φ = 2^{n}. It is clear that any such field will have property P N(n − 1). A well known example of such a field is the iterated power series field F = ℂ((X_{1}))((X_{2})) … ((X_{n})), where the supreme Pfister form is given by ⟨⟨X_{1}, …, X_{n}⟩⟩.

This also shows that for any n ≥ 2, there exist nonreal fields F with property P N(n) and u(F) = 2^{n+1}.

(ii) To get real fields with P N(n) (n ≥ 2) and u(F) = ũ(F) = 2^{n+1}, consider the real field K = ℚ(X_{1}, ⋯, X_{n}). Let π = ⟨⟨2, X_{1}, …, X_{n}⟩⟩. One readily sees that π is anisotropic and torsion (since ⟨⟨2⟩⟩ ≅ ⟨1, −2⟩ is torsion). Fix an ordering P ∈ X_{K}. Now consider

C = {field extensions L of K s.t. P extends to L and π_{L} anisotropic}

Clearly, K ∈ C, C is closed under direct limits, and if L ∈ C and L′ is a field with K ⊂ L′ ⊂ L, then L′ ∈ C. Then, by [B1, Theorem 6.1], there exists a field F ∈ C such that for any anisotropic form φ over F, dim φ ≥ 2, one has that F (φ) ∉ C. We claim that F has a unique ordering (which extends P), that F has P N(n) and that u(F) = ũ(F) = 2^{n+1}.

Now by construction, F is real with an ordering P′ extending P. Suppose there exists Q ∈ X_{F} with Q ≠ P′. Let a ∈ F such that a >_{P′} 0 and a <_{Q} 0, and consider q ≅ (2^{n+1} × ⟨1⟩) ⊥ ⟨−a⟩. Then q is anisotropic as it is positive definite at Q, and P′ (and thus P) extends to F (q) as q is indefinite at P′. However, since dim q = 2^{n+1} + 1 > 2^{n+1} = dim π, π stays anisotropic over F (q). Hence F (q) ∈ C, a contradiction. Thus, X_{F} = {P′}.

In particular, since π_{F} is torsion and anisotropic, we have u(F) ≥ 2^{n+1}. Suppose ũ(F) > 2^{n+1}. Then there exists an anisotropic t.i. form τ with dim τ > 2^{n+1}. A similar reasoning as above shows that F (τ) ∈ C, again a contradiction. Hence ũ(F) ≤ 2^{n+1} and we have u(F) = ũ(F) = 2^{n+1}.

Now let ψ be any form of dimension 2^{n} + 1 over F. If ψ is isotropic, it is a Pfister neighbor (Remark 6.6). So assume that ψ is anisotropic. Suppose first that ψ is t.i. and consider ρ = (π_{F} ⊥ −ψ)_{an}. Then 2^{n} − 1 ≤ dim ρ. If dim ρ > 2^{n} − 1 then dim ρ ≥ 2^{n} + 1 = dim ψ and | sgn_{P′} ρ| = | sgn_{P′} ψ| ≤ 2^{n} − 1, so in particular ρ is t.i. and thus P′ extends to F (ρ). Since we cannot have F (ρ) ∈ C, we must therefore have that π_{F (ρ)} is isotropic and hence hyperbolic, so ρ is similar to a subform of π_{F}. Thus, there exists x ∈ F^{∗} and a form γ, dim γ ≤ 2^{n} − 1 with xπ_{F} ≅ ρ ⊥ γ. Thus, in W F, we get xπ_{F} = π_{F} ⊥ −ψ ⊥ γ. But π_{F} ⊥ −xπ_{F} ∈ P_{n+2}F is torsion, therefore isotropic since u(F) = 2^{n+1} and thus hyperbolic (this actually shows that xπ_{F} ≅ π_{F} for any x ∈ F^{∗}). Hence, we have ψ = γ in W F with ψ anisotropic and dim ψ > dim γ, a contradiction. It then follows that dim ρ = 2^{n} − 1 and therefore π_{F} ≅ ρ ⊥ ψ, showing that ψ is a Pfister neighbor of π_{F}.

Now suppose that ψ is definite at the unique ordering P′ of F. After scaling, we may assume that ψ is positive definite. Let σ = 2^{n+1} × ⟨1⟩ ∈ P_{n+1}F. If ψ is a subform of σ then it is a Pfister neighbor and we are done. So suppose that ψ is not a subform of σ and let η ≅ (σ ⊥ −ψ)_{an}. We then have that dim η ≥ 2^{n} + 1 whereas sgn_{P′} η = 2^{n} − 1. In particular, η is t.i., and P′ extends to F (η). But F (η) ∉ C, so we must have that π_{F (η)} is isotropic and hence hyperbolic, and as above we have that π_{F} ≅ η ⊥ δ for some form δ with dim δ ≤ 2^{n} − 1. In W F, we thus get σ ⊥ −π_{F} = ψ ⊥ −δ ∈ I^{n+1}F. Now since dim ψ = 2^{n} + 1 ≥ dim δ + 2, we have that ψ ⊥ −δ is of dimension ≤ 2^{n+1} but not hyperbolic. By APH, we necessarily have that dim δ = 2^{n} − 1 and ψ ⊥ −δ ∈ GP_{n+1}F, so ψ is a Pfister neighbor, showing that F has property P N(n).

Let us finally remark that in this example, the proof shows that π_{F} is the unique anisotropic torsion (n+1)-fold Pfister form over F, and that there are two anisotropic (positive definite) (n + 1)-fold Pfister forms, namely σ and (σ ⊥ −π)_{an}. This also implies that I^{n+1}F/I^{n+2}F ≅ ℤ/2 × ℤ/2. □

We finish this paper with some remarks on a possible geometric interpretation of the property P N(n) which can be formulated in the language of Chow groups. We refer to [Kar], [EKM, §80].

Let φ be a (nondegenerate) quadratic form of dimension n + 2 ≥ 3, and let X = X_{φ} be the smooth projective n-dimensional quadric {φ = 0} over F. We call X (an)isotropic if φ is (an)isotropic. Let \overline{F} denote the algebraic closure of F and let \overline{X} = X\overline{F}. Let l_{0} be the class of a rational point in CH^{n}(\overline{X}), the Chow group of 0-dimensional cycles, and let 1 ∈ CH^{0}(X) be the class of X. A Rost correspondence on X is an element ρ ∈ CH^{n}(X × X) which, over \overline{F}, is equal to l_{0} × 1 + 1 × l_{0} ∈ CH^{n}(\overline{X} × \overline{X}). A Rost projector is a Rost correspondence that is also an idempotent in the ring of correspondences on X. It is known that if a quadric has a Rost correspondence, then it has in fact also a Rost projector (see [Kar, Rem. 1.4]). The study of Rost correspondences/projectors has proven to be crucial in the motivic theory of quadrics.

It is known that if X is isotropic, then l_{0} × 1 + 1 × l_{0} is actually the unique Rost projector on X (see [Kar, Lem. 5.1]). For anisotropic forms, the situation is much more complicated.

One knows the following (see [Kar, Prop. 6.2, 6.4]):

Theorem 6.16. Let φ be an anisotropic form over F of dimension ≥ 3, and let X = X_{φ}.

(i) If X possesses a Rost projector, then dim φ = 2^{n} + 1 for some n ≥ 1 (see Karpenko [Kar, Prop. 6.2, 6.4]).

(ii) If φ is a Pfister neighbor of dimension 2^{n} + 1 then X has a unique Rost projector (considered as element in CH^{r}(X × X), r = 2^{n} − 1) (see Izhboldin-Vishik [IV, Th. 1.12] for char(F) = 0, Elman-Karpenko-Merkurjev [EKM, Cor. 80.11] in the general case).

In view of part (i), it is natural to ask whether or not the converse of part (ii) also holds. This is still an open problem (see also [Kar, Conj. 1.6]):

Conjecture 6.17. If an anisotropic quadric X_{φ} possesses a Rost correspondence, then φ is a Pfister neighbor of dimension 2^{n} + 1 for some n ≥ 1.

Of course, by Theorem 6.16(ii), to prove the conjecture, one may assume that dim φ = 2^{n} + 1 for some n ≥ 1. Since 3-dimensional forms are always Pfister neighbors, trivially the conjecture holds in that case. The conjecture is also true in the cases n = 2, 3 as shown by Karpenko (see [Kar, Prop. 10.8, Th. 1.7]):

Theorem 6.18. Let φ be an anisotropic form over F of dimension 2^{n} + 1, n = 2, 3, and let X = X_{φ}. If X_{φ} possesses a Rost correspondence, then φ is a Pfister neighbor.

It is now natural to introduce the property RP (n) for n ≥ 1: RP (n): F has the property RP (n) for n ≥ 1 if every form φ over F of dimension

2^{n} + 1 has a Rost projector. In view of the above, we immediately get

Proposition 6.19. Let n ≥ 1.

(i) P N(n) implies RP (n).

(ii) If n ≤ 3, then RP (n) implies P N(n).

(iii) If Conjecture 6.17 holds, then RP (n) implies P N(n) for all n ∈ ℕ.

Conjecturally and in view of Theorem 1.1, we therefore get a ‘algebro-geometric’ criterion for the finiteness of the Hasse number:

Corollary 6.20. If Conjecture 6.17 holds, then ũ(F) < ∞ if and only if F has property RP (n) for some n ≥ 2.

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School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

  E-mail address: detlev.hoffmann@nottingham.ac.uk