TOTALLY DECOMPOSABLE QUADRATIC PAIRS

KARIM JOHANNES BECHER AND ANDREW DOLPHIN

Abstract. In this paper we show that a split central simple algebra with quadratic pair which decomposes into a tensor product of quaternion algebras with involution and a quaternion algebra with quadratic pair is adjoint to a quadratic Pfister form. This result is new in characteristic two, otherwise it is equivalent to the Pfister Factor Conjecture proven in [3]. Keywords: Central simple algebras, involutions, quadratic pairs, characteristic two, quadratic forms Mathematics Subject Classification (MSC 2010): 11E39, 11E81, 16W10, 16K20.

1. Introduction

To every symmetric or alternating bilinear form, and hence to every quadratic form when the base field is not of characteristic two, we can associate a central simple algebra with involution. In this way, the theory of quadratic forms is embedded into the larger theory of algebras with involution, and through the use of this correspondence, quadratic form theory has provided inspiration for the study of algebras with involution (see [12]). Pfister forms are a central concept in the modern algebraic theory of quadratic forms. It is therefore natural to look for class of central simple algebras with involution which extends the notion of a Pfister form. This question was first raised in [2]. Involutions adjoint to Pfister forms are tensor products of quaternion algebras with involution. Thus, tensor products of quaternion algebras with involution are a natural candidate.

In characteristic two, quadratic forms and symmetric bilinear forms are not equivalent objects. The relation between bilinear Pfister forms and totally decomposable involutions in characteristic two was studied in [7]. In order to have an object defined on a central simple algebra that corresponds to a quadratic form after splitting, the notion of a quadratic pair was introduced in [12, §5]. In particular, one may use quadratic pairs to give an intrinsic definition of twisted orthogonal groups in a manner that includes fields of characteristic two (see [12, §23.B]).

Algebras with quadratic pair associated to quadratic Pfister forms are tensor products of quaternion algebras with involution and a quaternion algebra with quadratic pair. One may ask whether all such totally decomposable quadratic pairs on a split central simple algebra are adjoint to a quadratic Pfister form. In characteristic different from two, where quadratic pairs are equivalent to orthogonal involutions, this is known to hold by the main result of [3], which says that in this case a totally decomposable orthogonal involution on a split algebra is adjoint to a Pfister form. In this article we prove the corresponding result for quadratic pairs over fields of characteristic two.

This work was supported by the Deutsche Forschungsgemeinschaft (project The Pfister Factor Conjecture in Characteristic Two, BE 2614/4), the FWO Odysseus programme (project Explicit Methods in Quadratic Form Theory), the Zukunftskolleg, Universität Konstanz and the Fonds Spéciaux de Recherche (postdoctoral funding) through the Université Catholique de Louvain.

Our approach is particular to fields of characteristic two. It allows us to capture more information on the resulting quadratic Pfister form (see Corollary 7.2), in a way that is not possible in general over fields of characteristic different from two (see Example 7.4). This approach uses the unusual properties of totally decomposable involutions in characteristic two found in [7]. In particular, we use that the isotropy behaviour of a totally decomposable orthogonal involution can be captured in the isotropy behaviour of an associated bilinear Pfister form (see Proposition 3.4).

The method used in [3] in characteristic different from two is based on a ramification exact sequence for Witt groups of quadratic forms over function fields of conics and the excellence property of these function fields. The latter result is known to extend to the case of arbitrary characteristic, but the former is not yet available in characteristic two in a suitable form. We intend to give a characteristic free proof in a future article. Our solution to the missing case of characteristic two involves several basic properties of quadratic pairs, which we did not find explicitly in the literature. Following [12, §5], we present such statements without assumptions on the characteristic, for ease of future reference.

2. Quadratic forms over fields

In this section we recall the terminology and results we use from quadratic form theory. We refer to [9, Chapters 1 and 2] as a general reference on symmetric bilinear and quadratic forms and for any basic notation and concepts not defined here. For two objects α and β in a certain category, we write α ≃ β to indicate that they are isomorphic, i.e. that there exists an isomorphism between them. This applies in particular to algebras with involution or with quadratic pair, but also to quadratic and bilinear forms, where the corresponding isomorphisms are called isometries. Throughout, let F be a field. Let char(F) denote the characteristic of F and let F^{×} denote the multiplicative group of F.

A bilinear form over F is a pair (V, b) where V is a finite-dimensional F -vector space and b is a F -bilinear map b : V × V → F. The radical of (V, b) is the set rad(V, b) = {x ∈ V | b(x, y) = 0 for all y ∈ V }. We say that (V, b) is degenerate if rad(V, b) ≠ {0}, and nondegenerate otherwise. Let φ = (V, b) be a bilinear form over F. We say that φ is symmetric if b(x, y) = b(y, x) for all x, y ∈ V. We call φ alternating if b(x, x) = 0 for all x ∈ V. The form φ is said to be isotropic if there exists an x ∈ V \{0} such that b(x, x) = 0, and anisotropic otherwise. We call a subspace W ⊆ V totally isotropic (with respect to b) if b|_{W ×W} = 0. If φ is nondegenerate and there exists a totally isotropic subspace W ⊆ V such that dim_{F} (W) = \frac{1}{2}dim_{F} (V), then we call φ metabolic.

For a_{1}, …, a_{n} ∈ F^{×} the symmetric F -bilinear map b : F^{n} × F^{n} → F given by (x, y) ↦ ∑_{i=1}^{n} a_{i}x_{i}y_{i} yields a symmetric bilinear form (F^{n}, b) over F, which we denote by ⟨a_{1}, …, a_{n}⟩. For a positive integer m, by an m-fold bilinear Pfister form over F we mean a nondegenerate symmetric bilinear form over F that is isometric to ⟨1, a_{1}⟩ ⊗ … ⊗ ⟨1, a_{m}⟩ for some a_{1}, …, a_{m} ∈ F^{×}. We call ⟨1⟩ the 0-fold bilinear Pfister form. By [9, (6.3)], a bilinear Pfister form is either anisotropic or metabolic.

By a quadratic form over F we mean a pair (V, q) of a finite-dimensional F -vector space V and a map q : V → F such that, firstly, q(λx) = λ^{2}q(x) holds for all x ∈ V and λ ∈ F, and secondly, the map b_{q} : V ×V → F , (x, y) ⟼ q(x+y)−q(x)−q(y) is F -bilinear. Then (V, b_{q}) is a symmetric bilinear form over F, called the polar form of (V, q). If b_{q} is nondegenerate, we say that (V, q) is nonsingular, otherwise we say that (V, q) is singular. If b_{q} is the zero map, then we say (V, q) is totally singular. By the quadratic radical of (V, q) we mean the set rad(V, q) = {x ∈ rad(V, b_{q}) | q(x) = 0}. We say that (V, q) is regular if rad(V, q) = {0}. For a symmetric bilinear form (V, b) over F, the map q : V → F given by q_{b}(x) = b(x, x) makes (V, q_{b}) a quadratic form over F. We call (V, q_{b}) the quadratic form associated to (V, b). If char(F) = 2 then this quadratic form is totally singular.

Let ρ = (V, q) and ρ′ = (V′, q′) be quadratic forms over F. By an isometry of quadratic forms ρ → ρ′ we mean an isomorphism of F -vector spaces f : V → V′ such that q(x) = q′(f(x)) for all x ∈ V. We say ρ is isotropic if q(x) = 0 for some x ∈ V \{0}, and anisotropic otherwise. By a totally isotropic subspace of ρ we mean an F -subspace W of V such that q|_{W} = 0. We call the maximum of the dimensions of all totally isotropic subspaces of ρ the Witt index of ρ, denoted i_{W} (ρ). Assume ρ is nonsingular. Then i_{W} (ρ) ⩽ \frac{1}{2}dim(ρ) (see [9, (7.28)]) and if i_{W} (ρ) = \frac{1}{2}dim(ρ) we say that ρ is hyperbolic. We denote the anisotropic part of ρ by ρ_{an} (see [9, (8.5)]).

We say that the quadratic forms ρ_{1} and ρ_{2} over F are similar if there exists an element c ∈ F^{×} such that ρ_{1} ≃ cρ_{2}. Recall the concept of a tensor product of a symmetric or alternating bilinear form and a quadratic form (see [9, p.51]). For a quadratic form ρ over F, we say that φ factors ρ if there exists a quadratic form ρ′ over F such that ρ ≃ φ ⊗ ρ′. Similarly, we say a quadratic form ρ′ over F factors ρ if there exists a symmetric bilinear form φ over F such that ρ ≃ φ ⊗ ρ′.

For a positive integer m, by an m-fold (quadratic) Pfister form over F we mean a quadratic form that is isometric to the tensor product of a 2-dimensional nonsingular quadratic form representing 1 and an (m − 1)-fold bilinear Pfister form over F. Pfister forms are either anisotropic or hyperbolic (see [9, (9.10)]).

Let ρ be a regular quadratic form over F. If dim(ρ) ⩾ 3 or if ρ is anisotropic and dim(ρ) = 2, then we call the function field of the projective quadric over F given by ρ the function field of ρ and denote it by F (ρ). In the remaining cases we set F (ρ) = F. This agrees with the definition in [9, §22]. For an anisotropic symmetric bilinear form φ, the quadratic form ρ associated to φ is regular. We call F (ρ) the function field of φ and we denote it by F (φ). Let K/F be a field extension. Then we write (V, q)_{K} = (V ⊗_{F} K, q_{K}) where q_{K} is the unique quadratic map such that q_{K}(v ⊗ k) = k^{2}q(v) for all v ∈ V and k ∈ K.

Proposition 2.1. Let ρ be a nonsingular quadratic form and let φ be an anisotropic bilinear Pfister form over F. If ρ_{F (φ)} is hyperbolic then either ρ is hyperbolic or φ factors ρ_{an}.

Proof. See [10, (5.2)]. □

3. Algebras with involution

We refer to [14] as a general reference on finite-dimensional algebras over fields, and for central simple algebras in particular, and to [12] for involutions. Let A be an (associative) F -algebra. We denote the centre of A by Z(A). For a field extension K/F, the K-algebra A ⊗_{F} K is denoted by A_{K}. An element e ∈ A is called an idempotent if e^{2} = e. An F -involution on A is an F -linear map σ : A → A such that σ(xy) = σ(y)σ(x) for all x, y ∈ A and σ^{2} = id_{A}.

Assume now that A is finite-dimensional and simple (i.e. it has no nontrivial twosided ideals). By Wedderburn’s Theorem (see [12, (1.1)]), A ≃ End_{D}(V) for a finitedimensional F -division algebra D and a finite-dimensional right D-vector space V. Furthermore, the centre of A is a field and dim_{Z(A)}(A) is a square number, whose positive square root is called the degree of A and is denoted deg(A). The degree of D is called the index of A and denoted ind(A). We call A split if ind(A) = 1, that is A ≃ End_{F} (V) for some finite-dimensional right F -vector space V. If Z(A) = F, then we call the F -algebra A central simple and we call a field extension K/F such that A_{K} is split a splitting field of A. If A is a central simple F -algebra then we denote Trd_{A} : A⟶F the reduced trace map and Nrd_{A} : A⟶F the reduced norm map, as defined in [12, (1.6)].

By an F -algebra with involution we mean a pair (A, σ) of a finite-dimensional central simple F -algebra A and an F -involution σ on A (note that we only consider involutions that are linear with respect to the centre of A, that is involutions of the first kind, here). We use the following notation: Sym(A, σ) = {a ∈ A | σ(a) = a}, Skew(A, σ) = {a ∈ A | σ(a) = −a} and Alt(A, σ) = {a − σ(a) | a ∈ A}. These are F -linear subspaces of A.

Let (A, σ) and (B, τ) be F -algebras with involution. By an isomorphism of F - algebras with involution Φ : (A, σ) → (B, τ) we mean an F -algebra isomorphism Φ : A → B satisfying Φ∘σ = τ ∘Φ. On the F -algebra A⊗_{F} B we obtain an F -involution σ ⊗ τ, whereby (A ⊗_{F} B, σ ⊗ τ) is an F -algebra with involution, which we denote by (A, σ) ⊗ (B, τ). For a field extension K/F we write (A, σ)_{K} = (A ⊗_{F} K, σ ⊗ id).

We call (A, σ) isotropic if there exists a ∈ A\{0} such that σ(a)a = 0, and anisotropic otherwise. An idempotent e ∈ A is called metabolic with respect to σ if σ(e)e = 0 and dim_{F} eA = \frac{1}{2}dim_{F} A. We call (A, σ) metabolic if A contains a metabolic idempotent element with respect to σ. For more information on metabolic involutions, see [6].

To every nondegenerate symmetric or alternating bilinear form φ = (V, b) over F we can associate an algebra with involution in the following way. Let A = End_{F} (V). Then there is a unique involution σ on A such that

b(x, f(y)) = b(σ(f)(x), y) for all x, y ∈ V and all f ∈ A.

We denote this F -involution on A by ad_{b}. We call (A, ad_{b}) the F -algebra with involution adjoint to φ and we denote it by Ad(φ). For every split F -algebra with involution (A, σ), there exists a nondegenerate symmetric or alternating bilinear form ψ over F such that (A, σ) ≃ Ad(ψ) (see [12, (2.1)]). The following is wellknown, but we include a proof for completeness.

Proposition 3.1. Let φ and ψ be nondegenerate symmetric bilinear forms over F. Then Ad(φ ⊗ ψ) ≃ Ad(φ) ⊗ Ad(ψ).

Proof. Let φ = (V, b) and ψ = (W, b′). Let f ∈ End_{F} (V) and g ∈ End_{F} (W). Then for all u, v ∈ V, w, t ∈ W we have

(b ⊗ b′)(f ⊗ g(u ⊗ w), (v ⊗ t)) = (b ⊗ b′)((f(u) ⊗ g(w)), (v ⊗ t))
= b(f(u), v) · b′(g(w), t)
= b(u, ad_{b}(f)(v)) · b′(w, ad_{b′}(g)(t))
= (b ⊗ b′)((u ⊗ w), (ad_{b}(f)(v) ⊗ ad_{b′}(g)(t))) .

Therefore, by bilinearity of b ⊗ b′, we have that ad_{b⊗b′}(f ⊗ g) = ad_{b}(f) ⊗ ad_{b}(g). Using this, it follows from the linearity of ad_{b⊗b′} that the natural isomorphism of F -algebras Φ : End_{F} (V) ⊗_{F} End_{F} (W) → End_{F} (V ⊗_{F} W) is an isomorphism of the F -algebras with involution in the statement. □

We distinguish two types of F -algebras with involution. An F -algebra with involution is said to be symplectic if it becomes adjoint to an alternating bilinear form over some splitting field of the associated F -algebra, and orthogonal otherwise. In characteristic different from two, these types are distinguished by the dimensions of the spaces of symmetric and alternating elements. However in characteristic two these dimensions do not depend on the type (see [12, (2.6)]).

An F -quaternion algebra is a central simple F -algebra of degree 2. Let Q be an F -quaternion algebra. By [12, (2.21)], the map Q → Q, x ↦ Trd_{Q}(x) − x is the unique symplectic involution on Q; it is called the canonical involution of Q. Any F -quaternion algebra has a basis (1, u, v, w) such that

u^{2} = u + a, v^{2} = b and w = uv = v − vu ,

for some a ∈ F with −4a ≠ 1 and b ∈ F^{×} (see [1, Chap. IX, Thm. 26]); such a basis is called an F -quaternion basis. Conversely, for a ∈ F with −4a ≠ 1 and b ∈ F^{×} the above relations uniquely determine an F -quaternion algebra (up to F -isomorphism), which we denote by [a, b). By the above, up to isomorphism any F -quaternion algebra is of this form. The following result can be recovered from [8, p.104, Thm. 4], but we include a direct argument.

Lemma 3.2. Let Q be an F -quaternion algebra and v ∈ Q\F be such that v^{2} ∈ F^{×}. There exist an element and u ∈ Q such that uv = v(1 − u) and u^{2} − u = a for some a ∈ F with −4a ≠ 1. That is, (1, u, v, uv) is an F -quaternion basis of Q.

Proof. If char(F) ≠ 2 then it is well-known that there exists an invertible element x ∈ Q such that xv + vx = 0 and we set u = x + \frac{1}{2}. Assume that char(F) = 2. Consider the F -linear map Q → Q, x ↦ xv + vx. Its kernel and its image are equal to F [v]. Hence uv + vu = v for some u ∈ Q, and it follows that u^{2} + u ∈ F [v] ∩ F [u] = F. □

With an F -quaternion basis (1, u, v, w) of Q, we define F -involutions γ, σ and τ on Q via their action on u and v as follows. We let

γ : u ↦ 1 − u, v ↦ −v
σ : u ↦ 1 − u, v ↦ v
τ : u ↦ u,  v ↦ −v .

Note that γ is the canonical involution on Q. Further, if char(F) = 2, then γ = σ and hence σ is symplectic. Otherwise σ is orthogonal. We use the notation

[a ·|· b) = (Q, γ), [a ·| b) = (Q, σ), [a |· b) = (Q, τ) .

Proposition 3.3. Let (Q, σ) be an F -quaternion algebra with orthogonal involution. Then there exists a ∈ F with −4a ≠ 1 and b ∈ F^{×} such that (Q, σ) ≃ [a |· b).

Proof. Let γ be the canonical involution of Q. By [12, (2.21)], we have σ = Int(v)∘γ for some invertible element v ∈ Skew(Q, γ)\F. Then v^{2} = −γ(v)v ∈ F^{×} and we set b = v^{2}. By Lemma 3.2, there exists an element u ∈ Q such that uv = v(1 − u) and u^{2} − u = a for some a ∈ F with −4a ≠ 1. Hence (1, u, v, uv) is an F -quaternion basis of Q and we have that γ(u) = 1 − u and γ(v) = −v. Hence σ(u) = u and σ(v) = −v, and further (Q, σ) ≃ [a |· b). □

We call an F -algebra with involution totally decomposable if it is isomorphic to a tensor product of F -quaternion algebras with involution. Note that by Proposition 3.1, the F -algebra with involution adjoint to a bilinear Pfister form over F is totally decomposable.

Let (A, σ) be an F -algebra with orthogonal involution. By [12, (7.1)], for any F - algebra with orthogonal involution (A, σ) with deg(A) even and any a, b ∈ Alt(A, σ) we have Nrd_{A}(a)F^{×2} = Nrd_{A}(b)F^{×2}. Therefore, as in [12, §7], we may make the following definition. The determinant of (A, σ), denoted Δ(A, σ), is the square class of the reduced norm of an arbitrary alternating unit, that is

Δ(A, σ) = Nrd_{A}(a) · F^{×2} in F^{×}/F^{×2} for any a ∈ Alt(A, σ) ∩ A^{×}.

For the rest of this section, we assume that char(F) = 2. Let (A, σ) be a totally decomposable F -algebra with orthogonal involution. That is

(A, σ) ≃ ⨂_{i=1}^{n}(Q_{i}, σ_{i})

where (Q_{i}, σ_{i}) are F -algebras with involution for i = 1, …, n. Note that we must have that (Q_{i}, σ_{i}) is orthogonal for all i = 1, …, n by [12, (2.23)]. Let d_{i} = Δ(Q_{i}, σ_{i}). Then the bilinear Pfister form π = ⟨⟨d_{1}, …, d_{n}⟩⟩ over F does not depend on the choice of the F -quaternion algebras with involution (Q_{i}, σ_{i}) in the decomposition of (A, σ) by [7, (7.3)]. We call this bilinear Pfister form the Pfister invariant of (A, σ) and denote it by 𝔓𝔣(A, σ). Note that by [7, (7.3)], for any field extension K/F we have that 𝔓𝔣((A, σ)_{K}) = (𝔓𝔣(A, σ))_{K}.

Proposition 3.4. Assume char(F) = 2. Let (A, σ) be a totally decomposable F – algebra with orthogonal involution. Then (A, σ) is anisotropic (resp. metabolic) if and only if 𝔓𝔣(A, σ) is anisotropic (resp. metabolic).

Proof. See [7, (7.5)]. □

4. Algebras with quadratic pair

We now recall the definition of and basic results we use on quadratic pairs. Let (A, σ) be an F -algebra with involution. We call an F -linear map f : Sym(A, σ) → F a semi-trace on (A, σ) if it satisfies f(x + σ(x)) = Trd_{A}(x) for all x ∈ A. By [4, (4.3)], if char(F) ≠ 2, then \frac{1}{2}Trd_{A}|_{Sym(A,σ)} is the unique semi-trace on (A, σ). On the other hand, if char(F) = 2, then the existence of a semi-trace on (A, σ) implies that Trd_{A}(Sym(A, σ)) = {0} and hence by [12, (2.6)] that (A, σ) is symplectic.

Given an element ℓ ∈ A with ℓ + σ(ℓ) = 1, the map f : Sym(A, σ) → F given by x ↦ Trd_{A}(ℓx) is a semi-trace on (A, σ), and conversely every semi-trace on (A, σ) is of this form by [12, (5.7)] (although the case where char(F) ≠ 2 and (A, σ) is symplectic is excluded there, the same proof applies). In this case, we say that the semi-trace f on (A, σ) is given by ℓ. For another element ℓ′ ∈ A such that ℓ′ + σ(ℓ′) = 1, we have that ℓ and ℓ′ give the same semi-trace on (A, σ) if and only if ℓ − ℓ′ ∈ Alt(A, σ) (see [12, (5.7)]).

An F -algebra with quadratic pair is a triple (A, σ, f) where (A, σ) is an F -algebra with involution, which is assumed to be orthogonal if char(F) ≠ 2 and symplectic if char(F) = 2, and where f is a semi-trace on (A, σ). In char(F) ≠ 2 the concept of an algebra with quadratic pair is equivalent to the concept of an algebra with orthogonal involution, as then the semi-trace given by \frac{1}{2} is the unique semi-trace on (A, σ).

Given two F -algebras with quadratic pair (A, σ, f) and (B, τ, g), by an isomorphism of F -algebras with quadratic pair Φ : (A, σ, f) → (B, τ, g) we mean an isomorphism of the underlying F -algebras with involution satisfying f = g ∘ Φ.

Let (A, σ, f) be an F -algebra with quadratic pair. We call (A, σ, f) isotropic if there exists an element s ∈ Sym(A, σ)\{0} such that s^{2} = 0 and f(s) = 0, and anisotropic otherwise. In particular, if (A, σ, f) is isotropic, then A has zero divisors. We call an idempotent e ∈ A hyperbolic with respect to σ and f if σ(e) = 1 − e and f(eA ∩ Sym(A, σ)) = {0}. We say that the F -algebra with quadratic pair (A, σ, f) is hyperbolic if A contains a hyperbolic idempotent with respect to σ and f.

We describe, following [12, §5], how a nonsingular quadratic form gives rise to an algebra with quadratic pair. Let ρ = (V, q) be a nonsingular quadratic form over F with polar form (V, b_{q}). By declaring

(v_{1} ⊗ w_{1}) ∗ (v_{2} ⊗ w_{2}) = b_{q}(w_{1}, v_{2}) · (v_{1} ⊗ w_{2}) for v_{1}, v_{2}, w_{1}, w_{2} ∈ V

a product ∗ is defined on the tensor product V ⊗_{F} V making it into an F -algebra. By declaring σ(v ⊗ w) = w ⊗ v for v, w ∈ V we obtain an F -involution σ on the F -algebra V ⊗_{F} V. Then by [12, (5.1)], the F -linear map Φ : V ⊗_{F} V → End_{F} (V) determined by

Φ(u ⊗ v)(w) = b_{q}(v, w)u for u, v, w ∈ V

yields an isomorphism of F -algebras with involution Ad(V, b_{q})⟶(V ⊗_{F} V, σ). According to [12, (5.11)] there is a unique semi-trace f_{q} : Sym(Ad(V, b_{q})) → F such that f_{q}(Φ(v ⊗ v)) = q(v) for v ∈ V, which yields an F -algebra with quadratic pair

Ad(ρ) = (End_{F} (V), ad_{b_{q}}, f_{q}) ,

called the adjoint F -algebra with quadratic pair of ρ. We say that an F -algebra with quadratic pair (A, σ, f) is adjoint to ρ if (A, σ, f) ≃ Ad(ρ). By [12, (5.11)], for any split F -algebra with quadratic pair (A, σ, f), there exists a nonsingular quadratic form ρ over F such that (A, σ, f) ≃ Ad(ρ), and for two nonsingular quadratic forms ρ and ρ′ over F, we have that Ad(ρ) ≃ Ad(ρ′) if and only if ρ and ρ′ are similar.

Proposition 4.1. Let ρ be a nonsingular quadratic form over F. Then ρ is isotropic (resp. hyperbolic) if and only if Ad(ρ) is isotropic (resp. hyperbolic).

Proof. The statement on isotropy follows from [12, (6.3) and (6.6)]. See [12, (6.13)] for the statement on hyperbolicity. □

For any field extension K/F we will use the notation (A, σ, f)_{K} for the K-algebra with quadratic pair (A_{K}, σ_{K}, f_{K}) where f_{K} : Sym(A_{K}, σ_{K})⟶K is the canonical extension of f to a K-linear map.

5. Tensor products of involutions and quadratic pairs

In this section we consider the tensor product of an algebra with involution with an algebra with quadratic pair. This corresponds to notion of the tensor product of a symmetric bilinear form and a quadratic form. We show that this tensor product is associative with the tensor product of algebras with involution. This property underlies our definition of a totally decomposable quadratic pair in the following section.

Proposition 5.1. Let (A, σ, f) be an F -algebra with quadratic pair and (B, τ) an F -algebra with involution. Then there is a unique semi-trace g on (B, τ) ⊗ (A, σ) such that g(s_{1} ⊗ s_{2}) = Trd_{B}(s_{1}) · f(s_{2}) for all s_{1} ∈ Sym(B, τ) and s_{2} ∈ Sym(A, σ).

Moreover, if the semi-trace f on (A, σ) is given by ℓ, then g is the semi-trace on (B, τ) ⊗ (A, σ) given by 1 ⊗ ℓ.

Proof. If char(F) ≠ 2 then this result is trivial. Assume that char(F) = 2. Note that as 1 ⊗ ℓ + (τ ⊗ σ)(1 ⊗ ℓ) = 1 ⊗ 1, the element 1 ⊗ ℓ gives a semi-trace on (B ⊗_{F} A, τ ⊗ σ) by [12, (5.7)]. For all s_{1} ∈ Sym(B, τ) and s_{2} ∈ Sym(A, σ) we have that

Trd_{B⊗_{F} A}((1 ⊗ ℓ)(s_{1} ⊗ s_{2})) = Trd_{B}(s_{1}) · Trd_{A}(ℓ · s_{2}) = Trd_{B}(s_{1}) · f(s_{2}) .

For the uniqueness statement, see [12, (5.18)]. □

Let (A, σ, f) be an F -algebra with quadratic pair and let (B, τ) be an F -algebra with involution, which is assumed to be orthogonal if char(F) ≠ 2. Then by [12, (2.23)], (B, τ) ⊗ (A, σ) is orthogonal if char(F) ≠ 2 and symplectic if char(F) = 2. We denote by (B, τ)⊗(A, σ, f) the F -algebra with quadratic pair (B ⊗_{F} A, τ ⊗σ, g), where g is the semi-trace g on (B, τ) ⊗ (A, σ) characterised in Proposition 5.1.

Proposition 5.2. Let φ be a symmetric bilinear form over F and ρ a nonsingular quadratic form over F. Then Ad(φ ⊗ ρ) ≃ Ad(φ) ⊗ Ad(ρ).

Proof. See [12, (5.19)]. □

Proposition 5.3. Let (B, τ) and (C, γ) be F -algebras with involution that are assumed to be orthogonal if char(F) ≠ 2 and let (A, σ, f) be an F -algebra with quadratic pair. Then

((B, τ) ⊗ (C, γ)) ⊗ (A, σ, f) ≃ (B, τ) ⊗ ((C, γ) ⊗ (A, σ, f)) .

Proof. Let Φ : (B ⊗_{F} C) ⊗_{F} A → B ⊗_{F} (C ⊗_{F} A) be the natural F -algebra isomorphism. Clearly Φ is compatible with the involutions in the statement. By [12, (5.7)], f is given by some ℓ ∈ A with ℓ + σ(ℓ) = 1. It follows from Proposition 5.1 that the semi-trace associated with ((B, τ) ⊗ (C, γ)) ⊗ (A, σ, f) is given by (1⊗1)⊗ℓ and the semi-trace associated with (B, τ)⊗((C, γ)⊗(A, σ, f)) is given by 1 ⊗ (1 ⊗ ℓ). It then easily follows that Φ is an isomorphism between the F -algebras with quadratic pair. □

By Proposition 5.3, the tensor product of two algebras with involution on the one hand, and the tensor product of an algebra with involution with an algebra with quadratic pair on the other hand, are mutually associative. That is, for F -algebras with involution (A, σ) and (B, τ) and an F -algebra with quadratic pair (C, γ, f), the expression (A, σ) ⊗ (B, τ) ⊗ (C, γ, f) is unambiguous.

Proposition 5.4. Assume char(F) = 2. Let (B, τ) and (C, σ) be F -algebras with symplectic involution. Then there exists a unique semi-trace h on (B, τ) ⊗ (C, σ) such that h(s_{1} ⊗ s_{2}) = 0 for all s_{1} ∈ Sym(B, τ) and s_{2} ∈ Sym(C, σ). Moreover, for any semi-trace f on (C, σ), h is the semi-trace associated with the F -algebra with quadratic pair (B, τ) ⊗ (C, σ, f).

Proof. For the existence and uniqueness of the semi-trace h, see [12, (5.20)]. Let f be any semi-trace on (C, σ). By [12, (5.7)], f is given by an element ℓ ∈ C with ℓ + σ(ℓ) = 1. Let g be the semi-trace associated with (B, τ) ⊗ (C, σ, f). Then by Proposition 5.1, g is given by 1 ⊗ ℓ. By [12, (2.6)], as (B, τ) is symplectic we have Trd_{B}(Sym(B, τ)) = {0}. Hence for all s_{1} ∈ Sym(B, τ) and s_{2} ∈ Sym(C, σ) we have

Trd_{B⊗_{F} C}((1 ⊗ ℓ)(s_{1} ⊗ s_{2})) = Trd_{B}(s_{1}) · Trd_{C}(ℓ · s_{2}) = 0 .

That is, g satisfies the characterising property of h in the statement. Therefore by the uniqueness of the semi-trace h, we have that g = h. □

Given two F -algebras with symplectic involution (B, τ) and (C, σ), we may define a semi-trace h on (B, τ) ⊗ (C, σ) in the following manner. If char(F) ≠ 2, then (B, τ)⊗(C, σ) is orthogonal by [12, (2.23)] and we let h = \frac{1}{2}Trd_{B⊗_{F} C}. If char(F) = 2, let h be the semi-trace on (B, τ) ⊗ (C, σ) characterised in Proposition 5.4. We denote the F -algebra with quadratic pair (B ⊗_{F} C, τ ⊗ σ, h) by (B, τ) ⊠ (C, σ). If char(F) = 2, then by Proposition 5.4 we have that (B, τ)⊠(C, σ) ≃ (B, τ)⊗(C, σ, f) for any choice of semi-trace f on (C, σ). In particular, by Proposition 5.3, for an F -algebra with symplectic involution (A, σ), the expression (A, σ) ⊗ (B, τ) ⊠ (C, γ) is unambiguous.

Hence given a tensor product of two F -algebras with symplectic involution, there is natural choice of a semi-trace making this product into a quadratic pair. We now consider this quadratic pair in the case where the F -algebras with involution are F -quaternion algebras with their canonical involutions. Let a ∈ F with −4a ≠ 1 and b ∈ F^{×} and let (Q, γ) = [a ·| b). Recall that this F -algebra with involution is orthogonal if char(F) ≠ 2 and symplectic if char(F) = 2. Let u ∈ Q be such that u^{2} = u + a and γ(u) = 1 − u and let f be the semi-trace on (Q, γ) given by u. Then we denote the F -algebra with quadratic pair (Q, γ, f) by [a || b).

Proposition 5.5. Let a, c ∈ F such that 4a ≠ −1 ≠ 4c and b, d ∈ F^{×}. Then

[a ·|· b) ⊠ [c ·|· d) ≃ [a + c + 4ac |· b) ⊗ [c || bd) .

Proof. Let (B, σ, f) = [a ·|· b)⊠[c ·|· d), (Q_{1}, γ_{1}) = [a ·|· b) and (Q_{2}, γ_{2}) = [c ·|· d). Let i, j ∈ Q_{1} be such that i^{2} = i + a, j^{2} = b and ij = j − ji and let u, v ∈ Q_{2} be such that u^{2} = u + c, v^{2} = d and uv = v − vu. In B we have that σ(i ⊗ 1) = 1 ⊗ 1 − i ⊗ 1, σ(j ⊗ 1) = −j ⊗ 1, σ(1 ⊗ u) = 1 ⊗ 1 − 1 ⊗ u and σ(1 ⊗ v) = −1 ⊗ v.

Let i′ = i ⊗ 1 + (1 − 2i) ⊗ u, j′ = j ⊗ 1, u′ = 1 ⊗ u and v′ = j ⊗ v. Then one easily checks that

Q′_{1} = F ⊕ F i′ ⊕ F j′ ⊕ F i′j′ and Q′_{2} = F ⊕ F u′ ⊕ F v′ ⊕ F u′v′

are σ-invariant F -subalgebras of B that commute elementwise with one another. We set τ_{1} = σ|_{Q′_{1}} and τ_{2} = σ|_{Q′_{2}}. We have

(Q′_{1}, τ_{1}) ≃ [a + c + 4ac |· b) and (Q′_{2}, τ_{2}) ≃ [c ·| bd) .

Hence (B, σ) ≃ [a + c + 4ac |· b) ⊗ [c ·| bd). If char(F) ≠ 2, then the semi-trace on (B, σ) is uniquely determined, and in this case there is nothing further to show.

Assume char(F) = 2. Then (B, σ, f) ≃ (Q_{1}, γ_{1}) ⊗ (Q_{2}, γ_{2}, h) for any choice of semi-trace h on (Q_{2}, γ_{2}) by Proposition 5.4. Let h to be the semi-trace given by u. Then for all s ∈ Sym(Q_{1} ⊗_{F} Q_{2}, γ_{1} ⊗ γ_{2}) = Sym(Q′_{1} ⊗_{F} Q′_{2}, τ_{1} ⊗ τ_{2}) we have that

Trd_{Q_{1}⊗_{F} Q_{2}}((1 ⊗ u) · s) = Trd_{Q_{1}⊗_{F} Q_{2}}(u′ · s) = Trd_{Q′_{1}⊗_{F} Q′_{2}}((1 ⊗ u′) · s) .

Hence (Q_{1}, γ_{1}) ⊗ (Q_{2}, γ_{2}, h) ≃ (Q′_{1}, τ_{1}) ⊗ (Q′_{2}, τ_{2}, g) for the semi-trace g on (Q′_{2}, τ_{2}) given by u′ by Proposition 5.1. That is, (B, σ, f) ≃ [a + c + 4ac |· b) ⊗ [c || bd). □

6. Totally decomposable quadratic pairs

We call an F -algebra with quadratic pair totally decomposable if it is isomorphic to a tensor product of a totally decomposable F -algebra with involution and an F -quaternion algebra with quadratic pair. It follows from Proposition 5.3 that taking the tensor product of a totally decomposable F -algebra with involution and a totally decomposable F -algebra with quadratic pair gives a totally decomposable algebra with quadratic pair.

Let (A, σ, f) be a totally decomposable F -algebra with quadratic pair. Then there exists a totally decomposable F -algebra with quadratic pair (B, τ) and an F - quaternion algebra with involution (Q, γ, g) such that (A, σ, f) ≃ (B, τ) ⊗ (Q, γ, h). If char(F) ≠ 2 then (B, τ) is necessarily orthogonal. We now show that, even if char(F) = 2, we may always find a decomposition as above where (B, τ) is orthogonal. This will allow us in the next section to use the Pfister invariant of (B, τ) to study the quadratic pair (A, σ, f).

Proposition 6.1. Let (A, σ, f) be a totally decomposable F -algebra with quadratic pair. Then there exists a totally decomposable F -algebra with orthogonal involution (B, τ) and an F -quaternion algebra with quadratic pair (Q, γ, g) such that (A, σ, f) ≃ (B, τ) ⊗ (Q, γ, g).

Proof. The result is trivial if char(F) ≠ 2. Assume that char(F) = 2. As (A, σ, f) is totally decomposable, there exist F -quaternion algebras with involution (Q_{i}, σ_{i}) for i = 1, …, n−1 and an F -quaternion algebra with quadratic pair (Q_{n}, γ, h) such that

(A, σ, f) ≃ (Q_{1}, σ_{1}) ⊗ … ⊗ (Q_{n−1}, σ_{n−1}) ⊗ (Q_{n}, γ, h) .

Suppose σ_{i} is symplectic. In particular, it is the canonical involution on Q_{i}, for some i ∈ {1, …, n − 1}. Then by Proposition 5.4, we have that

(Q_{i}, σ_{i}) ⊗ (Q_{n}, γ, h) ≃ (Q_{i}, σ_{i}) ⊠ (Q_{n}, γ) .

Hence, by Proposition 5.5, there exists an F -quaternion algebra with orthogonal involution (Q′_{i}, τ) and an F -quaternion algebra with quadratic pair (Q′_{n}, γ′, h′) such that

(Q_{i}, σ_{i}) ⊗ (Q_{n}, γ, h) ≃ (Q′_{i}, τ) ⊗ (Q′_{n}, γ′, h′) .

Using this argument repeatedly for all i = 1, …, n − 1 such that σ_{i} is symplectic, we modify our expression of (A, σ, f) above to obtain the result. □

For interest, we also record a characteristic two specific counterpart of the previous statement, which produces a symplectic instead of an orthogonal factor.

Proposition 6.2. Assume that char(F) = 2. Let (A, σ, f) be a totally decomposable F -algebra with quadratic pair with deg(A) = 2^{n}, where n ⩾ 2. Then there exist F - quaternion algebras with canonical involution (Q_{i}, γ_{i}) for i = 1, …, n such that (A, σ, f) ≃ (Q_{1}, γ_{1}) ⊗ … ⊗ (Q_{n−1}, γ_{n−1}) ⊠ (Q_{n}, γ_{n}) .

Proof. By Proposition 3.3, for every F -quaternion algebra with orthogonal involution (Q, τ), there exists an a ∈ F and b ∈ F^{×} such that (Q, τ) ≃ [a |· b). Similarly, by [4, (5.6)], for every F -quaternion algebra with quadratic pair (Q, γ, f) there exists an c ∈ F and d ∈ F^{×} such that (Q, γ, f) ≃ [c || d). The result thus follows using the isomorphism in Proposition 5.5 in a similar way as to how it is used in Proposition 6.1, but in the opposite direction. □

7. Totally decomposable quadratic pairs on a split algebra

We now prove our main result, that over fields of characteristic two a split algebra with totally decomposable quadratic pair is adjoint to a Pfister form. We use the following result, which is an approach unique to fields of characteristic two.

This approach gives more information on the m-fold Pfister form π adjoint to a totally decomposable quadratic pair on an algebra of degree 2^{m} over a field F of characteristic 2 after extending to splitting field K. Specifically, we show that we can always find an (m − 1)-fold bilinear Pfister form φ defined over F such that φ_{K} factors π.

Proposition 7.1. Assume char(F) = 2. Let (B, τ) be a totally decomposable orthogonal F -algebra with involution, (Q, γ, h) be an F -quaternion algebra with quadratic pair and (A, σ, f) = (B, τ) ⊗ (Q, γ, h). Then for any field extension K/F such that A_{K} is split, there exists a 1-fold Pfister form π over K such that (A, σ, f)_{K} ≃ Ad((𝔓𝔣(B, τ))_{K} ⊗ π).

Proof. Let φ = 𝔓𝔣(B, τ) and ρ a quadratic form over K with (A, σ, f)_{K} ≃ Ad(ρ). Note that dim(ρ) = 2dim(φ).

Assume first that (B, τ)_{K} is metabolic. Then by [5, (A.5)] we have that (A, σ, f)_{K} is hyperbolic and by Proposition 4.1 that ρ is hyperbolic. We may thus take π to be the hyperbolic 2-dimensional quadratic form.

Assume now that (B, τ)_{K} is not metabolic. Then the bilinear Pfister form φ_{K} is anisotropic by Proposition 3.4. We consider its function field L = K(φ_{K}). Since φ_{L} is metabolic, it follows by Proposition 3.4 that (B, τ)_{L} is metabolic. Hence, (A, σ, f)_{L} is hyperbolic by [5, (A.5)] and therefore ρ_{L} is hyperbolic by Proposition 4.1. By Proposition 2.1, there exists a non-trivial nonsingular quadratic form π′ over K such that ρ_{an} ≃ φ_{K} ⊗ π′. We have

dim(φ) · dim(π′) = dim(ρ_{an}) ⩽ dim(ρ) = 2dim(φ) .

As char(F) = 2, by [9, (7.32)] dim(π′) is even. It follows that dim(π′) = 2 and ρ_{an} ≃ ρ. In particular, π′ is similar to a 1-fold Pfister form π and ρ ≃ ρ_{an} ≃ φ_{K} ⊗π′. Hence ρ is similar to φ_{K} ⊗ π. □

Corollary 7.2. Assume that char(F) = 2. Let n ∈ ℕ, let (A, σ, f) be a totally decomposable F -algebra with quadratic pair with deg(A) = 2^{n+1} and let K/F be a field extension such that A_{K} is split. Then there exists an n-fold bilinear Pfister form φ over K and an (n + 1)-fold Pfister form ρ over K such that φ_{K} factors ρ and (A, σ, f)_{K} ≃ Ad(ρ).

Proof. For n = 0, this is trivial. Otherwise, by Proposition 6.1, there exists a totally decomposable F -algebra with orthogonal involution (B, τ) and an F -quaternion algebra with quadratic pair (Q, γ, h) such that (A, σ, f) ≃ (B, τ) ⊗ (Q, γ, h). The result then follows from Proposition 7.1 with φ = 𝔓𝔣(B, τ). □

Theorem 7.3. Let ρ be a nonsingular quadratic form over F with dim(ρ) ⩾ 2. Then Ad(ρ) is totally decomposable if and only if ρ is similar to a Pfister form.

Proof. If ρ is a Pfister form over F, then we can write ρ ≃ φ ⊗ π for a bilinear Pfister form φ and a 1-fold quadratic Pfister form π over F. Then by Proposition 5.2 we have that Ad(ρ) ≃ Ad(φ) ⊗ Ad(π). The F -algebra with involution Ad(φ) is totally decomposable by Proposition 3.1. As Ad(π) is an F -quaternion algebra with quadratic pair, it follows that Ad(ρ) is totally decomposable. Assume conversely that Ad(ρ) is totally decomposable. If char(F) ≠ 2, then any quadratic pair is equivalent to an orthogonal involution and thus the result corresponds to [3, Thm. 1]. If char(F) = 2, then ρ is similar to a Pfister form by Corollary 7.2. □

Let (A, σ, f) be a totally decomposable F -algebra with quadratic pair with deg(A) = 2^{m} and let K/F be a field extension such that A_{K} is split. Let π be the m-fold Pfister form over K such that (A, σ, f)_{K} ≃ Ad(π). In general, it is not possible to find an (m − 1)-fold quadratic Pfister form ρ over F such that ρ_{K} factors π. This is illustrated by the following example, which is a variation of an example in [2, (3.9)]. In particular, this example shows that Corollary 7.2 cannot be extended to cover fields of characteristic different from 2, where quadratic and bilinear Pfister forms are equivalent.

Example 7.4. Let n ∈ ℕ with n ⩾ 4. By [9, (38.4)] and its proof, there exists a field F such that all 3-fold quadratic Pfister forms over F are hyperbolic and there exist F -quaternion algebras Q_{1}, …, Q_{n} such that A = Q_{1} ⊗_{F} ⋯⊗_{F} Q_{n} is a division F -algebra. In particular, we have deg(A) = ind(A) = 2^{n}. For i = 1, …, n, let γ_{i} be the canonical involution on Q_{i} if char(F) = 2 and an orthogonal involution on Q_{i} if char(F) ≠ 2. We obtain a totally decomposable F -algebra with quadratic pair

(A, σ, f) = (Q_{1}, γ_{1}) ⊗ ⋯ ⊗ (Q_{n−1}, γ_{n−1}) ⊠ (Q_{n}, γ_{n}) .

By [11, (3.3) and §2.4] there exists a field extension K/F such that A_{K} is split and (A, σ, f)_{K} ≃ Ad(ρ) for some quadratic form ρ over K such that ind(A) divides i_{W} (ρ). As for any such ρ we have that dim(ρ) = 2^{n} = ind(A) and i_{W} (ρ) ⩽ \frac{1}{2}dim(ρ), it follows that i_{W} (ρ) = 0, that is, ρ is anisotropic. In particular, ρ is not factored by π_{K} for any (n − 1)-fold Pfister form π over F, as all such π are hyperbolic.

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E-mail address: KarimJohannes.Becher@uantwerpen.be, Andrew.Dolphin@uantwerpen.be

TOTALLY DECOMPOSABLE QUADRATIC PAIRS 13

Universiteit Antwerpen, Departement Wiskunde-Informatica, Middelheimlaan 1, 2020 Antwerpen, Belgium.