ORTHOGONAL PFISTER INVOLUTIONS IN CHARACTERISTIC TWO

ANDREW DOLPHIN

Abstract. We show that over a field of characteristic 2 a central simple algebra with orthogonal involution that decomposes into a product of quaternion algebras with involution is either anisotropic or metabolic. We use this to define an invariant of such orthogonal involutions in characteristic 2 that completely determines the isotropy behaviour of the involution. We also give an example of a non-totally decomposable algebra with orthogonal involution that becomes totally decomposable over every splitting field of the algebra. Keywords: Central simple algebras; quaternion algebras; involutions; Pfister forms; characteristic two, Pfister Factor Conjecture. Mathematics Subject Classification (MSC 2010): 11E39, 11E81, 12F05, 12F10.

1. Introduction

Over fields of characteristic different from 2 it is well-known that a quadratic form of dimension equal to a power of two is anisotropic or hyperbolic over any field extension if and only it is similar to a Pfister form. Using the correspondence between quadratic and bilinear forms over fields of characteristic different from 2, we can associate an orthogonal involution on a split central simple algebra to every quadratic form. It is then natural to consider whether there are central simple algebras with involution with analogous properties to Pfister forms. One potential type of algebra with involution to consider is the algebras with involution that are isomorphic to a product of quaternion algebras with involution. That is, totally decomposable algebras with involution.

Let (A, σ) be a central simple algebra of degree equal to a power of two with orthogonal involution over a field F. We denote the central simple algebra (resp. the algebra with involution) obtained by extending scalars over a field extension K/F as A_{K} (resp. (A, σ)_{K}).

Assuming that the characteristic of F is different from 2, in [2] it is asked whether the following are equivalent:

(1) (A, σ) is totally decomposable.

(2) For all field extensions K/F such that A_{K} is split, there exists a Pfister form π over K such that (A, σ)_{K} is isomorphic to the adjoint algebra with involution of π.

(3) For any field extension K/F, (A, σ)_{K} is either anisotropic or hyperbolic. The implication (1) ⇒ (2) is known as the Pfister Factor Conjecture, and was proven in [3]. The implication (1) ⇒ (3), and the equivalence (2) ⇔ (3), follows from the Pfister Factor Conjecture and the non-hyperbolic splitting result of [8]. The converse implication, (2) or (3) ⇒ (1), is still open in general.

Analogous questions may be asked when we consider fields of characteristic 2. Firstly, we may replace (A, σ) with a quadratic pair (see [11, Section 5]). The analogous result to the Pfister Factor Conjecture is shown in [6]. Otherwise the problem is open.

Alternatively we may formulate the question in terms of algebras with orthogonal involution and symmetric bilinear forms over a field of characteristic 2. The theory of symmetric bilinear forms in characteristic 2 has several features that mean we must be slightly more careful in our formulation of the analogous question to that posed in [2]. Over fields of characteristic different from 2, all 2-dimensional isotropic symmetric bilinear forms are isometric to the hyperbolic plane. This is not true in characteristic 2, and the wider variety of isotropic 2-dimensional forms means that we must use the weaker property of metabolicity rather than hyperbolicity in the formulation of our problem and be more restrictive with our statement. For example, there exist metabolic bilinear forms of dimension equal to a power of 2 that are not similar to bilinear Pfister forms (see (2.6)). That is, they are not isometric to a tensor product of 2-dimensional bilinear forms. Metabolicity for algebras with involution is studied in [4], and we recall the definitions and basic results that we use in Section 4.

Conversely however, the isotropy behaviour of symmetric bilinear forms over quadratic separable extensions is particularly simple (see (3.1)). We can often exploit this to investigate symmetric bilinear forms over fields of characteristic 2 with much simpler methods than those needed over fields of characteristic different from 2.

We therefore ask the following question, in analogy with the implication (1) ⇒ (3) above. Let F be a field of characteistic 2 and let (A, σ) be a totally decomposable F –algebra with orthogonal involution. For every field extension K/F, is (A, σ)_{K} either anisotropic or metabolic? In (5.2) we shall show that this question has a positive answer.

We shall also consider potential analogues of the other implications from the quesion asked in [2]. That a split totally decomposable F –algebra with orthogonal involution is adjoint to a bilinear Pfister form in charactersitic 2, an analogue of (1) ⇒ (2), can be shown with relatively simple arguments (see (6.7)). We shall give another slightly less direct proof of this result in (6.5) which allows us to determine the bilinear Pfister form in the statement explicitly. This result has also been independently shown using different methods in [15].

Natural analgoues of the equivalence (2) ⇔ (3) and the implication (2) ⇒ (1) do not hold in general in characteristic 2, and we discuss this in Section 7. In (8.4) we give an explicit example in characteristic 2 of an F –algebra with orthogonal involution that is not totally decomposable, but that becomes totally decomposable over every splitting field of the algebra.

2. Bilinear forms

In this section we recall the basic terminology and results we use from bilinear form theory. We refer to [7, Chapter 1] as a general reference on bilinear forms.

Let F be a field. Let char(F) denote the characteristic of F. A bilinear form over F is a pair (V, b) where V is a F –vector space and b is a F -bilinear map b : V × V → F. We say that a bilinear form (V, b) is symmetric if b(x, y) = b(y, x) for all x, y ∈ V. We call a bilinear form (V, b) alternating if b(x, x) = 0 for all x ∈ V. If (V, b) is an alternating form then we have that b(x, y) = −b(y, x) for all x, y ∈ V, that is, (V, b) is skew-symmetric. In particular every alternating form over a field of characteristic 2 is symmetric. We say (V, b) is nondegenerate if b(x, y) = 0 for all y ∈ V implies that x = 0.

An isometry of bilinear forms over F is a map ϕ : (V, b_{1}) → (W, b_{2}), where (V, b_{1}) and (W, b_{2}) are bilinear forms over F, such that ϕ : V → W is F –linear bijective F –vector space homomorphism and b_{1}(v, w) = b_{2}(ϕ(v_{1}), ϕ(v_{2})) for all v_{1}, v_{2} ∈ V. If such an isometry exists, we say (V, b_{1}) are (W, b_{2}) are isometric as bilinear forms and we write (V, b_{1}) ≃ (W, b_{2}). The orthogonal sum of the symmetric or alternating bilinear forms (V, b_{1}) and (W, b_{2}) is defined to be the map b : (V ⊕W)×(V ⊕W) → F given by b(v_{1} ⊕ w_{1}, v_{2} ⊕ w_{2}) = b_{1}(v_{1}, v_{2}) + b_{2}(w_{1}, w_{2}) for all v_{1}, v_{2} ∈ V and w_{1}, w_{2} ∈ W, and we write (V ⊕ W, b) = (V, b_{1}) ⊥ (W, b_{2}). The tensor product of the symmetric or alternating bilinear forms (V, b_{1}) and (W, b_{2}) is defined to be the map b′ : (V ⊗ W) × (V ⊗ W) → F given by b′(v_{1} ⊗ w_{1}, v_{2} ⊗ w_{2}) = b_{1}(v_{1}, v_{2}) · b_{2}(w_{1}, w_{2}) for all v_{1}, v_{2} ∈ V and w_{1}, w_{2} ∈ W and we write (V ⊗ W, b′) = (V, b_{1}) ⊗ (W, b_{2}).

We say a bilinear form (V, b) represents and element a ∈ F if there exists and x ∈ V \{0} such that b(x, x) = a. A bilinear form (V, b) is said to be isotropic if it represents 0, and anisotropic otherwise. Given a nondegenerate bilinear form (V, b) we call a subspace W ⊂ V totally isotropic (with respect to b) if b|_{W} = 0. We call (V, b) metabolic if it has a totally isotropic subspace W with dim_{F} (W) = \frac{1}{2}dim_{F} (V). Note that an alternating form is always metabolic. We say two bilinear forms (V, b) and (W, b′) are similar if there exists an a ∈ F^{×} such that (V, b) ≃ (W, ab′).

We now fix (V, b) to be a nondegenerate bilinear form over a field F for the rest of this section. Note that (V, b) can be decomposed as (V, b) ≃ (W_{1}, b_{1}) ⊥ (W_{2}, b_{2}) with (W_{1}, b_{1}) anisotropic and (W_{2}, b_{2}) metabolic. In this decomposition (W_{1}, b_{1}) is uniquely determined up to isometry (see [7, (1.27)]), whereas (W_{2}, b_{2}) is not in general. We call (W_{1}, b_{1}) the anisotropic part of (V, b), which we denote by (V, b)_{an}.

We denote ℍ = (F^{2}, h) where

h : F^{2} × F^{2} → F is given by (x, y) ↦ x^{t} ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) y

and call this the hyperbolic plane over F. We call a bilinear form (V, b) that is isometric to an orthogonal sum of hyperbolic planes hyperbolic. Over a field of characteristic 2, a bilinear form is hyperbolic if and only if it is alternating (see [7, (1.8)]).

For a_{1}, …, a_{n} ∈ F^{×} we denote by ⟨a_{1}, …, a_{n}⟩ the symmetric bilinear form (F^{n}, b) where

b : F^{n} × F^{n} → F is given by (x, y) ↦ ∑_{i=1}^{n} x_{i}a_{i}y_{i}.

We call such a form a diagonal form. A symmetric bilinear space that is isometric to a diagonal form is called diagonalisable. By [7, (1.17)], a nondegenerate symmetric bilinear form is diagonalisable if and only if it is not alternating. For a nondegenerate symmetric bilinear form φ over F we denote the metabolic bilinear form φ⊥φ by 𝕄(φ).

We frequently use the following isometry.

Lemma 2.1. Assume that char(F) = 2. Take a ∈ F^{×}. Then ⟨a⟩⊥ℍ ≃ ⟨a, a, a⟩.

Proposition 2.2. Assume that char(F) = 2. Let φ be a nondegenerate symmetric bilinear form over F. Then there exist n, m ∈ ℕ and elements a_{1}, …, a_{n} ∈ F^{×} such that

φ ≃ φ_{an}⊥𝕄(⟨a_{1}, …, a_{n}⟩)⊥m × ℍ

with the condition that φ_{an}⊥⟨a_{1}, …, a_{n}⟩ is anisotropic. Further φ_{an}⊥𝕄(⟨a_{1}, …, a_{n}⟩) is unique up to isometry and n is uniquely determined.

Proof. See [13, (2.1)]. □

We call the integer m in (2.2) the hyperbolicity index of φ.

For a ∈ F^{×} be denote the bilinear form ⟨1, −a⟩ over F by ⟨⟨a⟩⟩. For a_{1}, …, a_{n} ∈ F^{×}, we denote the ⟨⟨a_{1}⟩⟩ ⊗ … ⊗ ⟨⟨a_{n}⟩⟩ by ⟨⟨a_{1}, …, a_{n}⟩⟩. We call a bilinear form over F isometric to some ⟨⟨a_{1}, …, a_{n}⟩⟩ for some a_{1}, …, a_{n} ∈ F^{×}, an n-fold Pfister form.

Lemma 2.3. Let π be a Pfister form over F and a ∈ F^{×} an element represented by π. Then π ≃ aπ.

Proof. See [7, (6.2)]. □

The following result is well known, but we include a proof for convenience.

Corollary 2.4. Let φ be a symmetric bilinear form over F. If φ represents 1 and is similar to a Pfister form π then φ ≃ π.

Proof. Let a ∈ F be such that aφ ≃ π. Then since φ represents 1, we have that π represents a. Hence φ ≃ a^{2}φ ≃ aπ ≃ π by (2.3). □

Proposition 2.5. Let π be a nondegenerate symmetric bilinear form over F. If π is similar to a Pfister form then π_{K} is either anisotropic or metabolic for every field extension K/F. The converse holds if π is non-metabolic.

Proof. For the first statement see [7, (6.3)]. For the converse see [12, (5.5)] if char(F) = 2 and [7, (23.4)] otherwise. □

Lemma 2.6. Assume that char(F) = 2. Let φ be a nondegenerate metabolic symmetric bilinear form over F. Then φ is a Pfister form if and only if there exists and anisotropic Pfister form π such that

φ ≃ 𝕄(π)⊥n × ℍ ≃ π ⊗ ⟨⟨1, …, 1⟩⟩

for some integer n such that 2dim_{F} π + 2n = 2^{m} for some integer m.

Proof. For a Pfister form π over F, the isomorphism 𝕄(π)⊥n × ℍ ≃ π ⊗ ⟨⟨1, …, 1⟩⟩ follows from repeated use of (2.1). Therefore that φ is a Pfister form if φ ≃ 𝕄(π)⊥n × ℍ is clear. For the converse, see [1, (A.8)] and the comments following it. □

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

• q(λx) = λ^{2}φ(x) for all x ∈ V and λ ∈ F,

• b_{q} : V × V → F, (x, y) ⟼ q(x + y) − q(x) − q(y) is an bilinear form over F,

• if v ∈ V is such that q(v) = 0 and b_{q}(v, w) = 0 for all w ∈ V, then v = 0. We say φ is isotropic if q(x) = 0 for some x ∈ V \ {0} and anisotropic otherwise. For any bilinear form (V, b) over F we call the quadratic form consisting of the F –vector space V and a map q_{b} : V → F where q_{b}(x) = b(x, x) for all x ∈ V the associated quadratic form to (V, b). We denote this pair by { \atop (V, b)}.

Let (V, q) and (W, q′) be quadratic forms over F. We denote the orthogonal sum of (V, q) and (W, q′) by (V, q)⊥(W, q′). By an isometry of quadratic forms ϕ : (V, q) → (W, q′) we mean an isomorphism of F –vector spaces ϕ : V ⟶ W such that q = q′ ∘ ϕ. If such an isometry exists, we say (V, q) and (W, q′) are isometric and write (V, q) ≃ (W, q′).

3. Bilinear forms and separable extensions

In this section we recall some results on the behaviour of bilinear forms over separable extensions in characteristic 2 and derive some corollaries that we need.

Throughout, assume that char(F) = 2. Let K/F be a field extension. Then we write (V, b)_{K} = (V ⊗_{F} K, b_{K}) where b_{K} is the extension of b is a bilinear form on V ⊗_{F} K given by b_{K}(x ⊗ k_{1}, y ⊗ k_{2}) = k_{1}k_{2}b(x, y) for all x, y ∈ V and k_{1}, k_{2} ∈ K.

Proposition 3.1. Let L/F be a separable extension. Let φ be a nondegenerate symmetric bilinear form over F. Then φ_{L} is isotropic if and only if φ is isotropic.

Proof. See [9, (10.2.1)]. □

Corollary 3.2. Let L/F be a separable extension. Let φ and ψ be nondegenerate symmetric bilinear forms over F that are either anisotropic or metabolic. If φ_{L} ≃ ψ_{L} then φ ≃ ψ.

Proof. Assume that first that φ and ψ are anisotropic. Then (φ⊥ψ)_{L} is metabolic, and hence so is φ⊥ψ by (3.1) and therefore φ ≃ ψ by Witt Cancellation (see [7, (1.28)]).

Now assume that φ and ψ are metabolic. By (2.2) we have φ ≃ 𝕄(β_{1})⊥n×ℍ and ψ ≃ 𝕄(β_{2})⊥m × ℍ for anisotropic bilinear forms β_{1} and β_{2} over F and n, m ∈ ℕ. Since (β_{1})_{L} and (β_{2})_{L} are anisotropic by (3.1), it immediately follows from φ_{L} ≃ ψ_{L} that n = m by (2.2). Further, we have that 𝕄(β_{1})_{L} ≃ 𝕄(β_{2})_{L} by (2.2).

Let x = dim_{F} β_{1} = dim_{F} β_{2}. Using (2.1), adding β_{1} to each side of 𝕄(β_{1})_{L} ≃ 𝕄(β_{2})_{L} gives

(β_{1}⊥𝕄(β_{1}))_{L} ≃ (β_{1})_{L}⊥x × ℍ ≃ (β_{1})_{L}⊥𝕄(β_{2})_{L}.

If follows from (2.2) that (β_{1}⊥β_{2})_{L} is isotropic, and hence that (β_{1})_{L} and (β_{2})_{L} represent a comment element a ∈ L^{×}. That is (β_{1})_{L} ≃ ⟨a⟩⊥β′_{1} and β_{2} ≃ ⟨a⟩⊥β′_{2} for some anisotropic bilinear forms β′_{1} and β′_{2} over L. By (2.1) this gives

(β_{1}⊥𝕄(β_{2}))_{L} ≃ β′_{1}⊥⟨a⟩⊥𝕄(β′_{2}⊥⟨a⟩) ≃ β′_{1}⊥⟨a⟩⊥𝕄(β′_{2})⊥ℍ.

It follows from (2.2) that we can cancel hyperbolic planes, and hence

(β_{1})_{L}⊥(x − 1) × ℍ ≃ β′_{1}⊥𝕄(β′_{2}).

Inducting on x now gives that (β_{1}⊥β_{2})_{L} is metabolic. It follows that β_{1} ≃ β_{2} as in the anisotropic case above, and hence the result. □

Lemma 3.3. Let L/F be a separable field extension and φ be a symmetric bilinear form over F. If φ_{L} ≃ π, where π is a Pfister form over L, then φ is similar to a Pfister form over F.

Proof. First assume that φ is anisotropic and let be any field extension K/F such that φ_{K} is isotropic. Let K′ be a separable extension of K such that L is a subfield of K′. It follows that π_{K′} ≃ φ_{K′} and hence φ_{K′} is metabolic by (2.5). Therefore φ_{K} is metabolic by (3.1). Therefore φ is similar to a Pfister form by (2.5).

Now assume that φ is isotropic. Since φ is isotropic, it follows from (2.5) that π is metabolic. It then follows that φ is metabolic by (3.1). By (2.1) we have that π ≃ 𝕄(β)⊥n × ℍ for some natural number n and some anisotropic Pfister form β over L.

By (2.2), there exists an anisotropic symmetric bilinear form α and natural number m such that φ ≃ 𝕄(α)⊥m × ℍ. We may scale φ to assume that that α represents 1. Take a_{1}, …, a_{l} ∈ L^{×} such that α ≃ ⟨1, a_{1}, …, a_{l}⟩. It follows from (3.1) that α_{L} is anisotropic and the uniqueness of the decomposition in (2.2) implies that m = n and 𝕄(α_{L}) ≃ 𝕄(β). In particular dim_{F} (α) = dim_{F} (β).

It follows from (2.2) and [13, (2.6)] that α̃_{L} ≃ β̃. Consider the field L(α) = F (X_{1}, …, X_{l})(\sqrt{a_{1}X_{1}^{2} + … + a_{l}X_{l}^{2}}), where X_{1}, …, X_{l} are indeterminates. Then β_{L(α)} is isotropic as α̃_{L} ≃ β̃ and clearly α_{L(α)} is isotropic, and hence β_{L(α)} metabolic by (2.5). It then follows from [12, (5.3)] that aα_{L} ≃ β for some a ∈ L^{×} as dim_{F} (α) = dim_{F} (β). Since α represents 1, this implies that α_{L} ≃ β by (2.4) .

Hence α_{L} is isometric to a Pfister form, and hence α is similar to a Pfister form over F by the anisotropic case above. Since m = n and dim_{F} (α) = dim_{F} (β) it follows from (2.1) that φ ≃ ⟨⟨1, …, 1⟩⟩ ⊗ α. □

4. Algebras with involution

In this section we recall the basic definitions and results we use on central simple algebras with involution. We refer to [16] for a general reference on central simple algebras.

Let A be a finite-dimensional F –algebra. If A is simple (i.e. it has no nontrivial two sided ideals) and E is the centre of A, we can view A as an E–algebra and by Wedderburn’s Theorem (see [11, (1.1)]) we have that A ≃ End_{D}(V) for an F –division algebra D with centre E and a right D–vector space V. In this case dim_{E}(A) is a square, and the positive root of this integer is called the degree of A and is denoted deg(A). The degree of D is called the index of A and is denoted ind(A). We call A split if ind(A) = 1. If E = F, then we call the F –algebra A central simple. An F –quaternion algebra is a central simple F -algebra of degree 2. For any field extension K/F we will use the notation A_{K} = A ⊗_{F} K. We call a field extension K/F a splitting field of A if A_{K} is split.

Lemma 4.1. If F is separably closed, then all central simple F –algebras are split.

Proof. See [5, (9.2)]. □

For an F –algebra A and b ∈ A^{×} we denote by Int(b) the inner automorphism A → A given by c ↦ bcb^{−1}.

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}. We call an F –involution of the first kind if F = E, the centre of A. We do not consider F –involutions of the second kind here (see [11, Section 2.B] for more details on this kind of involution). By an F –algebra with involution we mean a pair (A, σ) of a finite-dimensional F –algebra A and an F –involution σ on A of the first kind.

A homomorphism of F –algebras with involution is a map φ : (A, σ) → (B, τ), where (A, σ) are (B, τ) F –algebras with involution, such that φ : A → B is an F – algebra homomorphism and φ∘σ = τ∘φ; if φ is bijective then this is an isomorphism. If an isomorphism φ : (A, σ) → (B, τ) exists, then we say we say (A, σ) and (B, τ) are isomorphic as F –algebras with involution and we write (A, σ) ≃ (B, τ). For any field extension K/F we will use the notations σ_{K} = σ ⊗ id_{K} and (A, σ)_{K} = (A_{K}, σ_{K}).

We call an F -algebra with involution totally decomposable if is it isomorphic to the tensor product of F –quaternion algebras with involution.

To every nondegenerate symmetric or alternating bilinear form (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 call (A, σ) the adjoint involution to (V, b) and we denote it by Ad(V, b). Moreover, for every split F –algebra with involution (A, σ), there exists a nondegenerate symmetric or alternating bilinear form (V, b) such that Ad(V, b) ≃ (A, σ) (see [11, (2.1)]).

Proposition 4.2. Let (V, b) and (V′, b′) be nondegenerate symmetric bilinear forms over F. Then Ad(V, b) ≃ Ad(V′, b′) if and only if there exists an λ ∈ F^{×} such that (V, λb) ≃ (V′, b′).

Proof. See [11, (4.2)]. □

Let (A, σ) be an F –algebra with involution. Then it is well known (see [11, (2.1)]) that in the case where the algebra A is split, that is A ≅ End_{F} (V) for some F -vector space V, each F -involution on A is adjoint to a nondegenerate symmetric or alternating bilinear space on V. An F –algebra with involution of the first kind is said to be symplectic if it becomes adjoint to an alternating bilinear form over a splitting field of the F –algebra, and orthogonal otherwise. Note that this does not depend on the choice of splitting field (see [11, (2.5)]).

Let (A, σ) be an F –algebra with involution. We call (A, σ) isotropic if there exists a ∈ A\{0} such that σ(a)a = 0, and anisotropic otherwise. We call an element e ∈ A idempotent if e^{2} = e. 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; by [4, (4.3)], we may substitute the condition dim_{F} eA = \frac{1}{2}dim_{F} A for the condition that (1−e)(1−σ(e)) = 0 in this definition. We call (A, σ) metabolic if A contains a metabolic idempotent element with respect to σ.

Proposition 4.3. Let (V, b) be a nondegenerate alternating or symmetric bilinear form over F. Then (V, b) is isotropic (resp. metabolic) if and only if Ad(V, b) is isotropic (resp. metabolic).

Proof. See [4, (4.8)]. □

Note that, in particular, it follows from the definition of symplectic involutions that any split F –algebra with involutions (A, σ) is isomorphic to Ad(n × ℍ) for some integer n.

Let (A, σ) be an F –algebra with involution. Let

Sym(A, σ) = {a ∈ A | σ(a) = a} and Alt(A, σ) = {a − σ(a) | a ∈ A}.

These are F –linear subspaces of A.

An F –algebra with involution (B, τ) is called a part of (A, σ) if there exists an idempotent e ∈ Sym(A, σ) and an F –algebra isomorphism φ : B → eAe such that φ∘τ = σ∘φ. In this case, we say that the idempotent e ∈ Sym(A, σ) defines the part (B, τ). Let (B, τ) be a part of of (A, σ) defined by an idempotent e ∈ Sym(A, σ). Then we call the part of (A, σ) defined by the idempotent 1 − e ∈ Sym(A, σ) a counterpart of (B, τ) in (A, σ) and we say that (A, σ) is an orthogonal sum of (B, τ) and its counterpart. Note however that the counterpart of a part (B, τ) of an algebra with involution is not uniquely determined by (B, τ) (see [5, (4.2)])

Parts of F –algebras with involution correspond to subforms of hermitian forms whose adjoint involution is isomorphic to those F –algebras with involution (see [11, (4.2)]). We state this more precisely for the split case.

Lemma 4.4. Let φ be a nondegenerate symmetric or alternating bilinear form over F. An F –algebra with involution (B, τ) is a part of Ad(φ) if and only if there exist bilinear forms φ′ and φ′′ over F such that φ′⊥φ′′ ≃ φ and (B, τ) ≃ Ad(φ′). Let e ∈ Sym(A, σ) be the idempotent that defines (B, τ) in (A, σ). Then the idempotent (1 − e) ∈ Sym(A, σ) defines a part of Ad(φ) that is isomorphic to Ad(φ′′).

Proof. See [5, (4.1)]. □

If char(F) ≠ 2, then a part of an F –algebra with involution (A, σ) must be of the same type as (A, σ). This is not the case if char(F) = 2 (see [5, (4.6)]).

Lemma 4.5. Assume that char(F) = 2. Let (A, σ) be an orthogonal F –algebra with involution. Let (B, τ) be a part of (A, σ). Then if (B, τ) is symplectic, any counterpart of (B, τ) is orthogonal.

Proof. Let K be a separable closure of F and let (C, γ) be a counterpart of (B, τ). Then A_{K}, B_{K} and C_{K} are split by (4.1) and hence there exists a bilinear form φ over F such that (A, σ)_{K} ≃ Ad(φ). By (4.4) there exists forms φ_{1} and φ_{2} such that φ ≃ φ_{1}⊥φ_{2} and Ad(φ_{1}) ≃ (B, τ)_{K} and Ad(φ_{2}) ≃ (C, γ)_{K}. By definition, (B, τ) symplectic implies that φ_{1} is hyperbolic. Since (A, σ) is orthogonal φ is not hyperbolic, and therefore we must have that φ_{2} is not hyperbolic. Hence (C, γ) is not symplectic. □

We call a part (B, τ) of (A, σ) the anisotropic part of (A, σ) if (B, τ) is direct and its counterpart is metabolic. This part is unique up to isomorphism by [4, (3.6)] and we denote it by (A, σ)_{an}.

Assume that char(F) = 2. We say (A, σ) is direct if for all a ∈ A such that σ(a)a ∈ Alt(A, σ), it follows that a = 0. If A is split, then an algebra with involution is direct if and only if it is anisotropic (see [5, (7.1)]). We call a part (C, γ) of (A, σ)_{an} the direct part of (A, σ) if (C, γ) is direct and its counterpart in (A, σ)_{an} is symplectic. This part is unique up to isomorphism by [5, (7.5)], as is its counterpart, which we call the even part of (A, σ), denoted (A, σ)_{ev}.

Lemma 4.6. Assume that char(F) = 2. Let L be the separable closure of F. Let (A, σ) be an F –algebra with involution and let φ be a nondegenerate symmetric bilinear form over L such that (A, σ)_{L} ≃ Ad(φ). If (A, σ) has a non-trivial even part, then the hyperbolicity index of φ is strictly positive.

Proof. This follows from (4.4) since split algebras with symplectic involutions are adjoint to hyperbolic bilinear forms by definition. □

In characteristic 2, the direct part of an F –algebra with involution classifies the isotropy behaviour of the involution when extending scalars to a separable extension of F that splits the algebra.

Proposition 4.7. Assume that char(F) = 2. Let (A, σ) be an F –algebra with involution and let L/F be a separable algebraic field extension such that A_{L} is split. Then ((A, σ)_{L})_{an} ≃ ((A, σ)_{dir})_{L}.

Proof. See [5, (9.3)]. □

5. Isotropy of totally decomposable orthogonal involutions

In this section we prove our main result. That a totally decomposable orthogonal involution is anisotropic or metabolic over every field extension. We assume throughout this section that char(F) = 2.

Lemma 5.1. Let (A, σ) be a totally decomposable orthogonal F –algebra with involution. Then (A, σ) is anisotropic if and only if it is direct.

Proof. That (A, σ) is anisotropic if it is direct is clear as 0 ∈ Alt(A, σ).

Assume that (A, σ) is anisotropic. Over the separable closure L of F all F – quaternion algebras are split by (4.1). Therefore (A, σ)_{L} isomorphic to Ad(π) for some Pfister form π over L. Hence by (4.3) and (2.5), (A, σ)_{L} is either metabolic or anisotropic. If (A, σ)_{L} is anisotropic, then (A, σ) is direct by (4.7).

Otherwise, (A, σ)_{an} is symplectic by (4.7). However (A, σ) is orthogonal, and hence in this case must have a non-trivial metabolic part. □

Theorem 5.2. Let (A, σ) be an orthogonal F –algebra with involution such that (A, σ) ≃ ⨂_{i=1}^{n}(Q_{i}, σ_{i}), where (Q_{i}, σ_{i}) are F –algebras with involution for all i ∈ {1, …, n}. If (A, σ) is isotropic then it is metabolic.

Proof. By [11, (2.23)] we have that the (Q_{i}, σ_{i}) are all orthogonal as (A, σ) is orthogonal. We proceed by induction. The result is trivial for n = 1. Assume that it is true for n − 1.

Let (A, σ) ≃ (B, τ) ⊗ (Q_{n}, σ_{n}) be an isotropic orthogonal F –algebra with involution such that (Q_{i}, σ_{i}) are orthogonal F –algebras with involution for i = 1, …, n and (B, τ) = ⨂_{i=1}^{n−1}(Q_{i}, σ_{i}). By the inductive assumption, we may assume that (B, τ) and (Q_{n}, σ_{n}) are anisotropic, as if one of them is metabolic then (A, σ) is metabolic. In particular (B, τ) is direct by (5.1).

Let L be the separable closure of F. By (2.5), (A, σ)_{L} is metabolic and hence by [5, (9.4)], (A, σ) is some orthogonal sum of an anisotropic symplectic F –algebra with involution (C, γ) and a metabolic F –algebra with involution. If (C, γ) is trivial, then we are done.

By (4.1) all L–algebras are split, hence we may find π and ψ ≃ ⟨1, a⟩, Pfister forms over L such that Ad(π) ≃ (B, τ) and Ad(ψ) ≃ (Q_{n}, σ_{n}). By (4.7), (B, τ)_{L} is anisotropic and hence so is π. Since (A, σ)_{L} ≃ Ad(π ⊗ ψ) is metabolic, we have

π ⊗ ψ ≃ 𝕄(β)⊥n × ℍ

for some anisotropic bilinear form β over L. Hence π⊥aπ is metabolic, and in particular it represents 0. As π is anisotropic, it follows that π represents a and therefore aπ ≃ π by (2.3). Hence π ⊗ ψ ≃ 𝕄(π) and we must have that 𝕄(π) ≃ 𝕄(β) by (2.2). In particular, we have that n = 0. Hence (C, γ) is trivial by (4.6), as required. □

Lemma 5.3. A totally decomposable orthogonal F -algebra with involution is either direct or metabolic. In particular, the even part of a totally decomposable orthogonal F –algebra with involution is trivial.

Proof. If an orthogonal F -algebra with involution is anisotropic then it is direct by (5.1). Otherwise it is metabolic by (5.2) and has trivial anisotropic part. □

Corollary 5.4. Let L be the separable closure of F. A totally decomposable orthogonal F -algebra with involution (A, σ) is metabolic if and only if (A, σ)_{L} is metabolic. Otherwise (A, σ) is anisotropic.

Proof. This follows immediately from (5.3) and (4.7). □

Question 5.5. Let (A, σ) be a non-metabolic F –algebra with involution. Assume that for every field extension K/F we have (A, σ)_{K} is either anisotropic or metabolic. Is (A, σ) totally decomposable?

6. An invariant of totally decomposable algebras

In this section we use (5.2) to define an invariant on totally decomposable involutions and give an application of this result.

Let (A, σ) be an F –algebra with orthogonal involution. As in [11, Section 7] the determinant of (A, σ), denoted Δ(A, σ), is the square class of the reduced norm of any alternating unit. That is

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

This does not depend on the choice of a ∈ Alt(A, σ) ∩ A^{×} (see [11, (7.1)]).

Lemma 6.1. Let (Q, σ) be a split orthogonal F –quaternion algebra with involution and let d ∈ F be a representative of the class of Δ(Q, σ) in F^{×}/F^{×2}. Then (Q, σ) ≃ Ad(⟨1, b⟩).

Proof. This follows from [11, (7.3), (3)] and the fact that the determinant of a 2–dimensional bilinear form classifies it up to similarity. □

We now assume throughout the rest of this section that char(F) = 2.

Lemma 6.2. Let (A, σ) be an orthogonal F –algebra with involution such that (A, σ) ≃ ⨂_{i=1}^{n}(Q_{i}, σ_{i}) ≃ ⨂_{i=1}^{n}(Q′_{i}, σ′_{i}) where (Q_{i}, σ_{i}) and (Q′_{i}, σ′_{i}) are orthogonal F –algebras with involution for all i ∈ {1, …, n}. Let Δ_{i} = Δ(Q_{i}, σ_{i}) and Δ′_{i} = Δ(Q′_{i}, σ′_{i}). Then ⟨⟨Δ_{1}, …, Δ_{n}⟩⟩ ≃ ⟨⟨Δ′_{1}, …, Δ′_{n}⟩⟩.

Proof. Let π = ⟨⟨Δ_{1}, …, Δ_{n}⟩⟩ and π′ = ⟨⟨Δ′_{1}, …, Δ′_{n}⟩⟩. Let L be the separable closure of F. Since (Q_{i})_{L} and (Q′_{i})_{L} are split for all i ∈ {1, …, n} by (4.1), it follows from (6.1) that (A, σ) ≃ Ad(π) ≃ Ad(π′). Therefore by (4.2) there exists a λ ∈ L such that λπ_{L} ≃ π′_{L}. However, since π and π′ are Pfister forms, it follows from (2.4) that π_{L} ≃ π′_{L}. That π ≃ π′ now follows from (3.2). □

Theorem 6.3. Let (Q_{i}, σ_{i}) be orthogonal F –algebras with involution for all i ∈ {1, …, n} and let Δ_{i} = Δ(Q_{i}, σ_{i}). Then the map that associates ⟨⟨Δ_{1}, …, Δ_{n}⟩⟩ to the F –algebra with involution ⨂_{i=1}^{n}(Q_{i}, σ_{i}) induces a map from the set of isomorphism classes of totally decomposable orthogonal F –algebras with involution of degree 2^{n} to the set of isometry classes of n–fold Pfister forms over F. This map is compatible with scalar extension.

Proof. That the map induced by associating ⟨⟨Δ_{1}, …, Δ_{n}⟩⟩ with ⨂_{i=1}^{n}(Q_{i}, σ_{i}) is well defined follows directly from (6.2). Let K/F be a field extension and let π be the Pfister form over K associated with (A, σ)_{K} via the map in the statement. That π ≃ ⟨⟨Δ_{1}, …, Δ_{n}⟩⟩_{K} follows using the same argument as in (6.2). □

With notation as in (6.3), the Pfister form ⟨⟨Δ_{1}, …, Δ_{n}⟩⟩ associated to a totally decomposable F –algebra with orthogonal involution (A, σ) is uniquely determined by (A, σ) up to isometry. We call this Pfister form the Pfister invariant of (A, σ). We denote it by 𝔓𝔣(A, σ).

Question 6.4. For totally decomposable orthogonal F –algebras with involution (A, σ) and (A, τ), does 𝔓𝔣(A, σ) ≃ 𝔓𝔣(A, τ) imply that (A, σ) ≃ (A, τ)?

Corollary 6.5. Let (A, σ) be a totally decomposable orthogonal F –algebra with involution. Then

(1) (A, σ) is metabolic if and only if 𝔓𝔣(A, σ) is metabolic.

(2) If 𝔓𝔣(A, σ) is not metabolic, then (A, σ) is anisotropic.

(3) For any field K such that A_{k} is split, (A, σ)_{K} ≃ Ad(𝔓𝔣(A, σ)_{K}).

Proof. Let L be the separable closure of F. Then we clearly have that (A, σ)_{L} ≃ Ad(𝔓𝔣(A, σ)_{L}). Hence (A, σ)_{L} is anisotropic (resp. metabolic) if and only if Ad(𝔓𝔣(A, σ)_{L}) is anisotropic (resp. metabolic) by (4.3). This is equivalent to (1) and (2) by (3.1) and (5.4).

Now let φ be a symmetric bilinear form over K such that (A, σ)_{K} ≃ Ad(φ). By scaling we may assume that φ represents 1. Then by (4.2), there exists a λ ∈ L^{×} such that λφ_{L} ≃ π_{L}. Note that this implies that φ_{L} is either anisotropic or metabolic and hence so is φ by (3.1). Then φ_{L} ≃ π_{L} by (2.4). That φ ≃ π_{K} now follows from (3.2). □

Remark 6.6. Let k be a field of characteristic different from 2. It is shown in [2, (3.9)] that in general one cannot associate a bilinear form over k to any F –algebra with involution (A, σ) so that φ shares its anisotropy behaviour with (A, σ) as the Pfister invariant is shown to do in (6.5). Moreover, they show that for a field extension K/k such that A_{K} is split, there does not always exist a bilinear form φ over k such that (A, σ)_{K} ≃ Ad(φ_{K}). This gives (3).

Remark 6.7. Part (2) of (6.5) can be thought of as one characteristic 2 version of the Pfister Factor Conjecture (see [3]). That is, any totally decomposable orthogonal involution over a split algebra is adjoint to a Pfister form.

This can be shown more directly as follows. Let (A, σ) be a totally decomposable F –algebra with orthogonal involution such that A is split. Then (A, σ) ≃ Ad(φ) for some bilinear form φ over F which we may assume represents 1. Let L be a separable closure of F. Then, since every L-quaternion algebra is split by (4.1), we must have that (A, σ)_{L} is isomorphic to a product of split L–quaternion algebras with involution, and hence (A, σ)_{L} ≃ Ad(π) for some Pfister form π over L. That φ_{L} ≃ π now follows from (4.2) and (2.4), and hence φ is a Pfister form by (3.3) and (2.4).

However (6.5) not only shows that these split totally decomposable algebras with involution are adjoint to Pfister forms, but gives an explicit description of the Pfister forms. This result was also independently obtained in [15, (4.6)] with different methods that do not rely on the non-split behaviour of totally decomposable involutions.

Corollary 6.8. Let (Q_{1}, σ_{1}) and (Q_{2}, σ_{2}) be orthogonal F –quaternion algebras with involution such that Δ(Q_{1}, σ_{1}) = Δ(Q_{2}, σ_{2}). Then (Q_{1}, σ_{1}) ⊗ (Q_{2}, σ_{2}) is metabolic.

Proof. It is clear that 𝔓𝔣((Q_{1}, σ_{1})⊗(Q_{2}, σ_{2})) is metabolic, and hence so is (Q_{1}, σ_{1})⊗ (Q_{2}, σ_{2}) by (6.5). □

Remark 6.9. Let k be a field of arbitrary characteristic. Let (Q_{1}, σ_{1}) and (Q_{2}, σ_{2}) be split orthogonal k–quaternion algebras with involution such that Δ = Δ(Q_{1}, σ_{1}) = −Δ(Q_{2}, σ_{2}). Then by (6.1) we have that (Q_{1}, σ_{1}) ≃ Ad(⟨1, Δ⟩) and (Q_{2}, σ_{2}) ≃ Ad(⟨1, −Δ⟩). Moreover, since we have (Q_{1}, σ_{1}) ⊗ (Q_{2}, σ_{2}) ≃ Ad(⟨1, Δ⟩ ⊗ ⟨1, −Δ⟩) it is clear that (Q_{1}, σ_{1}) ⊗ (Q_{2}, σ_{2}) is metabolic.

(6.8) shows that this behaviour extends to the non-split case when char(k) = 2. Note that is is not the case when char(k) ≠ 2, where Δ(Q_{1}, σ_{1}) = −Δ(Q_{2}, σ_{2}) does not even imply that Q_{1} ⊗ Q_{2} is not division in general, as (6.11) shows.

Let k be a field with char(k) ≠ 2. For a, b ∈ k^{×}, let (1, u, v, w) be basis elements of a k-vector space such that u^{2} = a, v^{2} = b and w = uv = −vu. Then (1, u, v, w) generate an k–quaternion algebra which we denote (a, b)_{k}. There is a unique symplectic k–involution γ on (a, b)_{k} given by γ(u) = −u and γ(v) = −v (see [11, (2.21)]). We call this k–involution the canonical involution on (a, b)_{F}.

Lemma 6.10. Let k be a field with char(k) ≠ 2. Let K = k(X, Y, Z) where X, Y and Z are indeterminates. Then we have

(X, Y)_{K} ⊗ (−1, Z)_{K} ≃ (X, Y Z)_{K} ⊗ (−X, Z)_{K}.

Proof. Let φ ≃ ⟨−X, −Y, XY, 1, −Z, Z⟩ and ψ ≃ ⟨−X, −Y Z, XY Z, X, −Z, −XZ⟩. Direct computation shows that φ⊥(−ψ) ≃ ⟨⟨X, Y, Z⟩⟩⊥2 × ℍ. By [14, Chapter XII, (2.13)], it then follows that (X, Y)_{K} ⊗ (−1, Z)_{K} ≃ (X, Y Z)_{K} ⊗ (−X, Z)_{K}. □

Example 6.11. Let k be a field with char(k) ≠ 2. Further assume that −1 ∉ k^{2}. Let K = k(X, Y, Z) where X, Y and Z are indeterminates. Then the K–algebra A = (X, Y)_{K} ⊗ (−1, Z)_{K} is division by [16, (19.6, Corollary c)].

By (6.10) we have A ≃ (X, Y Z)_{K}⊗(−X, Z)_{K}. Let (1, u_{1}, v_{1}, w_{1}) and (1, u_{2}, v_{2}, w_{2}) be K–bases of Q_{1} = (X, Y Z)_{K} and Q_{2} = (−X, Z)_{K} respectively. Let γ_{1} and γ_{2} be the canonical involutions of Q_{1} and Q_{2} respectively. Then (Q_{1}, Int(u_{1}) ∘ γ_{1}) and (Q_{2}, Int(u_{2})∘γ_{2}) are orthogonal K–algebras with involution by [11, (2.21)]. Further by [11, (7.4)] Δ(Q_{1}, Int(i_{1}) ∘ γ_{1}) = X and Δ(Q_{2}, Int(i_{2}) ∘ γ_{2}) = −X.

However, every F –involution on A must be anisotropic, as for all a ∈ A^{×} we have σ(a)a ≠ 0 since A is division. In particular, (Q_{1}, Int(i_{1}) ∘ γ_{1}) ⊗ (Q_{2}, Int(i_{2}) ∘ γ_{2}) is not metabolic.

7. Involutions adjoint to Pfister forms after splitting

In this section we show that a non-metabolic algebra with involution that is anisotropic or metabolic over every field extension is adjoint to a Pfister form over any splitting field, but that the converse fails in general. This result also shows that over fields of characteristic 2, there exist non-totally decomposable algebras with involution that become totally decomposable over some field extension. We construct an explicit example in Section 8. We assume throughout this section that char(F) = 2.

Proposition 7.1. Let (A, σ) be a split anisotropic orthogonal F –algebra with involution. Assume that for all field extensions K/F, (A, σ)_{K} is either anisotropic or metabolic. Then there exists a Pfister form over F such that (A, σ) ≃ Ad(π).

Proof. By hypothesis (A, σ) must be anisotropic. Let φ be a bilinear form over F such that (A, σ) ≃ Ad(φ). We may assume φ represents 1. Then φ is anisotropic and φ_{K} is either anisotropic or metabolic over every field extension K/F by (4.3). Hence φ is a Pfister form by (2.5) and (2.4). □

Proposition 7.2. Let (A, σ) be an anisotropic F –algebra with involution. Assume that for all field extensions K/F, (A, σ)_{K} is either anisotropic or metabolic. Let L be the separable closure of F. Then (A, σ)_{L} is anisotropic and there exists a Pfister form π over L such that (A, σ)_{L} ≃ Ad(π).

Proof. By the hypothesis, (A, σ) is anisotropic and not symplectic. If (A, σ)_{an} = (A, σ) is not isomorphic to (A, σ)_{dir} then it follows from (4.6) and (4.7) that (A, σ)_{L} is neither anisotropic nor metabolic. Hence (A, σ) is direct. Therefore, by (4.7), (A, σ)_{L} is anisotropic and by the hypothesis, either anisotropic or metabolic over every field extension of L. Therefore, there exists a Pfister form π over L such that (A, σ)_{L} ≃ Ad(π) by (7.1). □

Theorem 7.3. Let (A, σ) be an anisotropic orthogonal F –algebra with involution. Assume that for all field extensions K′/F, (A, σ)_{K′} is either anisotropic or metabolic. Then for every field extension K/F such that A_{K} is split, there exists a Pfister form over K such that (A, σ)_{K} ≃ Ad(π).

Proof. Let L be a separable closure of F. By (7.2) there exists a Pfister form π over L such that (A, σ)_{L} ≃ Ad(π).

Let K/F be a field extension such that A_{K} is split, and let ψ be a bilinear form over K such that (A, σ)_{K} ≃ Ad(ψ). We may assume that ψ represents 1. Let K′ be a separable extension of K such that L is a subfield of K′. It follows that (A, σ)_{K′} ≃ Ad(φ_{K′}) and hence π_{K′} ≃ ψ_{K′} by (4.2) and (2.4). Hence ψ is similar to a Pfister form by (2.5). □

Question 7.4. In the situation of (7.3), can we find a Pfister form π′ over F such that π′_{K} ≃ π?

Note that the converse to (7.3) does not hold in general as we now show.

Lemma 7.5. Let (A, σ) be an orthogonal F –algebra with involution with deg(A) = 2^{n} for some integer n such that (A, σ)_{an} ≃ (A, σ)_{ev} and the counterpart of (A, σ)_{an} is totally decomposable. Let K/F be a field extension such that A_{K} is split. Then there exists a Pfister form over K such that (A, σ)_{K} ≃ Ad(π).

Proof. Denote the counterpart of (A, σ)_{an} by (B, τ). Note that (B, τ) must be orthogonal by (4.5). Since A_{K} is split there exists some symmetric bilinear form φ over K such that (A, σ)_{K} ≃ Ad(φ).

Then by (4.4) it follows from the hypothesis that φ ≃ 𝕄(π)⊥n × ℍ for some bilinear form π over K and integer n with 2dim_{F} (π) + 2n = 2^{m} for some integer m such that ((A, σ)_{ev})_{K} ≃ Ad(n × ℍ) and (B, τ)_{K} ≃ Ad(𝕄(π)). Since (B, τ) is totally decomposable, it follows from (6.5) that π is a Pfister form and hence so is φ by (2.6). □

Proposition 7.6. Let (A, σ) be an orthogonal F –algebra with involution with deg(A) = 2^{n} for some integer n such that (A, σ)_{an} ≃ (A, σ)_{ev} is non-trivial and a counterpart of (A, σ)_{an} is totally decomposable and non-trivial. Then the following hold:

(a) For any field extension K/F such that A_{K} is split (A, σ)_{K} is totally decomposable.

(b) (A, σ) is isotropic but not metabolic.

(c) (A, σ) is not totally decomposable.

Proof. Statement (a) follows directly from (7.5). Statement (b) is clear as (A, σ)_{an} is non-trivial but not isomorphic to (A, σ). It is also clear that (A, σ) is neither direct nor metabolic, hence (5.3) gives (c). □

8. A non-totally decomposable involution that becomes totally decomposable after splitting

We now use hermitian forms to construct an explicit example of the type of algebra with involution described in (7.6). Throughout this section, assume that char(F) = 2.

Let (D, θ) be an F –division algebra with involution of the first kind. A hermitian form over (D, θ) is a pair (V, h) where V is a finite-dimensional right D-vector space and h is a non-degenerate bi-additive map h : V × V → D such that

h(x, yd) = h(x, y)d and h(y, x) = θ(h(x, y))

holds for all x, y ∈ V and d ∈ D. We say (V, h) is non-degenerate if h(x, y) = 0 for all y ∈ V implies that x = 0. We say (V, h) represents an element a ∈ D if h(x, x) = a for some x ∈ V \{0}. We call a hermitian form (V, h) isotropic it represents 0, and anisotropic otherwise. We call a hermitian form (V, h) metabolic if there exists a subspace W ⊂ V such that h|_{W} = 0 and dim_{F} (W) = \frac{1}{2}dim_{F} (V).

For a_{1}, …, a_{n} ∈ D^{×} ∩ Sym(D, θ), we denote by ⟨a_{1}, …, a_{n}⟩_{θ} the hermitian form (D^{n}, h) where

h : D^{n} × D^{n} → D, is given by (x, y) ↦ ∑_{i=1}^{n} θ(x_{i})a_{i}y_{i}.

There is a well known correspondence between non-degenerate hermitian forms on V and F –involutions on A, generalising the correspondence between bilinear forms and involutions on a split algebra:

Proposition 8.1. Let (D, θ) be an F –division algebra with involution, V a right D–vector space and let A = End_{D}(V). For every non-degenerate hermitian form (V, h), there is a unique F –involution σ on A such that

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

Proof. See [11, (4.1)]. □

In the situation of (8.1), we call (A, σ) the F –algebra with involution adjoint to (V, h) and we write Ad(V, h) = (End_{D}(V), σ).

Proposition 8.2. Let (V, h) be a nondegenerate hermitian form over and F – division algebra with involution (D, θ). Then (V, h) is isotropic (resp. metabolic) if and only if Ad(V, h) is isotropic (resp. metabolic).

Proof. See [4, (4.8)]. □

Let a ∈ F be such that −4a ≠ 1 and let b ∈ F^{×}. Let K = F (α) where α^{2}+α = a and let τ be the non-trivial F –automorphism of K. Let (1, u, v, w) be basis elements of the F -vector space K ⊕ vK such that u^{2} = u + a, v^{2} = b and w = uv = τ(u). Then (1, u, v, w) generate an F –quaternion algebra which we denote by [a, b)_{F}.

Let Q = [a, b)_{F} for a ∈ F and b ∈ F^{×} as above. Let γ be the F –involution on Q given by γ(u) = τ(u) and γ(v) = v. This is the unique symplectic involution on Q (see [11, (2.21)]). Let Alt(Q, γ) = {γ(s) + s | s ∈ Q}. Then direct computation gives Alt(Q, γ) = F and that for s = t_{1} + t_{2}u + t_{3}v + t_{4}w where t_{1}, …, t_{4} ∈ F we have

γ(s)s = t_{1}^{2} + t_{1}t_{2} + t_{2}^{2}a + (t_{3}^{2} + t_{3}t_{4} + t_{4}^{2}a)b.

If we consider Q as a 4-dimensional F –vector space, Q together with the map n_{Q} : Q → F, s ↦ γ(s)s for s ∈ Q can be considered as a quadratic form over F.

Lemma 8.3. Let k be a field of arbitrary characteristic and let F = k(X, Y), where X, Y are indeterminates. Let Q = [X, Y)_{F}. Then Q is an F –division algebra.

Proof. By [7, (12.5)], Q is division if and only if (Q, n_{Q}) is anisotropic. By [7, (17.4)], (Q, n_{Q}) is anisotropic if and only if

ρ = (k(Y)^{2}, (t_{1}, t_{2}) ↦ t_{1}^{2} + t_{1}t_{2} + t_{2}^{2}Y)

is anisotropic. By [7, (17.3)], ρ is anisotropic if and only if there exists an element a ∈ k[Y] such that 1 + a + a^{2}Y = 0.

Assume a = a_{n}Y^{n} + … + a_{1}Y + a_{0} for n > 0 and a_{0}, …, a_{n} ∈ k with a_{n} ≠ 0. Substituting this expression for a into 1+a+a^{2}Y = 0 gives a_{n}^{2}Y^{2n+1} +b = 0 where b ∈ k[Y] such that deg_{Y} (b) < 2n + 1. This contradicts a_{n} ≠ 0. Therefore we must have that deg_{Y} (a) = 0. However, since Y ∉ k, it follows that a = 0, and hence ρ is anisotropic. □

Example 8.4. Let k be a field of characteristic 2 and let F = k(X, Y, Z) where X, Y and Z are indeterminates. Let Q = [X, Y)_{F}, which is an F –divison algebra by (8.3).

Let ψ be the hermitian form ⟨1, Z, v, v⟩_{γ} over (Q, γ) and let (A, σ) ≃ Ad(ψ). We have that Ad(⟨1, Z⟩_{γ}) is symplectic and that Ad(⟨v, v⟩_{γ}) and (A, σ) are orthogonal by [11, (4.2)].

Since ⟨v, v⟩_{γ} is clearly metabolic, by (8.2) we have that Ad(⟨v, v⟩_{γ}) is metabolic. We have that ⟨1, Z⟩_{γ} is isotropic if and only if γ(a_{1})a_{1} + Zγ(a_{2})a_{2} = 0 for some (a_{1}, a_{2}) ≠ (0, 0) ∈ Q × Q. That is, if and only if (Q, n_{Q})⊥Z(Q, n_{Q}) is isotropic. That (Q, n_{Q})⊥Z(Q, n_{Q}) is anisotropic follows from [7, (17.14)] using an argument similar to that in (8.3). Hence ⟨1, Z⟩_{γ} is anisotropic, and hence so is Ad(⟨1, Z⟩_{γ}) by (8.2). It follows that ψ is not metabolic by [10, Chapter 1, (6.1.1)] and hence (A, σ) is not metabolic by (8.2).

Finally, Ad(⟨v, v⟩_{γ}) ≃ Ad(⟨1, 1⟩) ⊗ Ad(⟨v⟩_{γ}) and Ad(⟨v⟩_{γ}) ≃ (Q, σ′) for some orthogonal F –involution σ′. That is, Ad(⟨v, v⟩_{γ}) is totally decomposable. Therefore (A, σ) satisfies the conditions of (7.6). That is, (A, σ) is a non-totally decomposable orthogonal F –algebra with involution such that (A, σ)_{L} is totally decomposable for any field extension L/F such that A_{L} is split.

Question 8.5. Let (A, σ) be an orthogonal F –algebra with involution. Assume that there exists a field extension L/F such that (A, σ)_{L} is anisotropic and A_{L} is split. That is, by (4.7), (A, σ) is direct. Assume further that for all field extensions K/L, (A, σ)_{K} is either anisotropic or metabolic. Does it follow that for every field extension K′/F, (A, σ)_{K′} is either anisotropic or metabolic?

Acknowledgements

I am indebted to Karim Johannes Becher and Jean-Pierre Tignol for their input and feedback regarding a preliminary version of this article.

This work was supported by the Deutsche Forschungsgemeinschaft (project The Pfister Factor Conjecture in characteristic two, BE 2614/4) through Universität Konstanz and by the Fonds speciaux de recherche (postdoctoral funding).

References

[1] J. Arason and R Baeza. Relations in I^{n} and I^{n}W_{q} in characteristic 2. J. Algebra, 314:895–911, 2007.

[2] E. Bayer-Fluckieger, R. Parimala, and A. Quéguiner-Mathieu. Pfister involutions. Proc. Indian Acad. Sci. Math. Sci., 113:365–377, 2003.

[3] K. J. Becher. A proof of the Pfister Factor Conjecture. Invent. Math., 173(1):1–6, 2008.

[4] A. Dolphin. Metabolic involutions. Journal of Algebra, 336(1):286–300, 2011.

[5] A. Dolphin. Decomposition of algebras with involution in characteristic 2. Journal of Pure and Applied Algebra, 217(9):1620–1633, 2013.

[6] A. Dolphin. The Pfister Factor Conjecture for quadratic pairs. Preprint, 2013.

[7] R. Elman, N. Karpenko, and A. Merkurjev. The Algebraic and Geometric Theory Quadratic Forms, volume 56 of Colloq. Publ., Am. Math. Soc. Am. Math. Soc., 2008.

[8] N. A. Karpenko. Hyperbolicity of orthogonal involutions. Doc. Math., Extra volume: Andrei A. Suslin sixtieth birthday:371–392, 2010.

[9] M. Knebusch. Grothendieck und Wittringe von nichtausgearteten symmetrischen Bilinearformen. Math-natw. Klasse 3. Sitzb. Heidelberg Akademie Wiss., 1969/1970.

[10] M.-A. Knus. Quadratic and Hermitian Forms over Rings, volume 294 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, 1991.

[11] M.-A. Knus, A.S. Merkurjev, M. Rost, and J.-P. Tignol. The Book of Involutions, volume 44 of Colloq. Publ., Am. Math. Soc. Am. Math. Soc., 1998.

[12] A. Laghribi. Witt kernels of function field extensions in characteristic 2. J. Pure Appl. Algebra, 199:167–182, 2005.

[13] A. Laghribi and P. Mammone. Hyper-isotropy of bilinear forms in characteristic 2. Contemporary Mathematics, 493:249–269, 2009.

[14] T. Y. Lam. Introduction to Quadratic Forms over Fields, volume 67 of Grad. Studies in Math. Amer. Math. Soc., 2004.

[15] M.G. Mahmoudi and A.-H. Nokhodkar. On split products of quaternion algebras with invo-

lution. Preprint, http://arxiv.org/abs/1306.2598, 2013.

[16] R. Pierce. Associative Algebras. Graduate texts in mathematics. Springer-Verlag, 1982.

ICTEAM Institute, Université catholique de Louvain, Louvain-la-Neuve, Belgium

E-mail address: andrew.dolphin@uclouvain.be