INVOLUTIONS AND STABLE SUBALGEBRAS

KARIM JOHANNES BECHER, NICOLAS GRENIER-BOLEY, AND JEAN-PIERRE TIGNOL

Abstract. Given a central simple algebra with involution over an arbitrary field, étale subalgebras contained in the space of symmetric elements are investigated. The method emphasizes the similarities between the various types of involutions and privileges a unified treatment for all characteristics whenever possible. As a consequence a conceptual proof of a theorem of Rowen is obtained, which asserts that every division algebra of exponent two and degree eight contains a maximal subfield that is a triquadratic extension of the centre. Keywords: Central simple algebra, Double Centraliser Theorem, maximal étale subalgebra, capacity, Jordan algebra, crossed product, characteristic two Classification (MSC 2010): 16H05, 16R50, 16W10

1. Introduction

We investigate étale algebras in the space of symmetric elements of a central simple algebra with involution over an arbitrary field, emphasizing the similarities between the various types of involutions and avoiding restrictions on the characteristic. In Section 2 and Section 3 we recall the terminology and some crucial techniques for algebras with involution. We enhance this terminology in a way that allows us to avoid unnecessary case distinctions in the sequel, according to the different types of involution and to the characteristic. To this end we introduce in Section 3 the notion of capacity of an algebra with involution. It is defined to be the degree of the algebra if the involution is orthogonal or unitary, and half the degree if the involution is symplectic. In Section 5 we isolate a notion of neat subalgebra, which captures the features of separable field extensions of the centre consisting of symmetric elements while avoiding the pathologies that may arise with arbitrary étale algebras. We prove their existence and determine their maximal dimension to be equal to the capacity (Theorem 4.1 and Proposition 5.6). In Section 6, given a neat quadratic subalgebra K, we establish the existence of a neat subalgebra L linearly disjoint from K and centralising K and such that the composite KL is a neat algebra of maximal dimension (Theorem 6.10). In Section 7 we apply this result to construct neat biquadratic subalgebras in the space of symmetric elements of central simple algebras of degree 4 with orthogonal or unitary involutions, and similarly of central simple algebras of degree 8 with symplectic involutions (Theorem 7.4). As a consequence, we obtain a conceptual proof of a theorem of Rowen, which asserts that division algebras of exponent 2 and degree 8 are elementary abelian crossed products, i.e., they contain a maximal subfield which is a triquadratic separable extension of the centre (Corollary 7.7). Actually we obtain directly a refined version of this result which says that any symplectic involution on a central simple algebra of degree 8 stabilizes some triquadratic étale extension of the centre (Theorem 7.6). This has been proven in [7, Lemma 6.1] for division algebras in characteristic different from two, but there the proof uses Rowen’s Theorem, which we obtain here as a consequence. This illustrates the usefulness of involutions in the investigation of central simple algebras of exponent two.

Date: October 2017.

The results of this paper will be used in [4], which proposes a common approach to the definition of the first cohomological invariant (discriminant) of the involutions of capacity four of various types through Pfister forms in arbitrary characteristic.

2. Algebras

In this preliminary section we introduce and recall some definitions and facts from the theory of finite-dimensional simple and semisimple algebras. Our standard references are [11] and [8].

Let F be an arbitrary field. For a commutative F -algebra K we set [K : F] = dim_{F} K. Recall that an F -algebra is étale if it is isomorphic to a finite product of finite separable field extensions of F. An étale F -algebra is said to be split if it is F -isomorphic to F^{n} for some n ∈ ℕ.

2.1. Lemma. Let L be a split étale F -algebra with |F | > [L : F]. Then L = F [a] for an element a ∈ L^{×} which is separable over F.

Proof. In F^{n} any element a = (a_{1}, …, a_{n}) with distinct a_{1}, …, a_{n} ∈ F^{×} is invertible and has minimal polynomial ∏_{i=1}^{n}(X − a_{i}) over F, which is separable of degree n, whereby L = F [a]. □

Let A be an F -algebra. We denote by Z(A) the centre of A and by A^{op} the opposite algebra of A.

2.2. Lemma. Let K = Z(A) and assume that K is an étale F -algebra. Let L be a commutative semisimple F -subalgebra of A which is F -linearly disjoint from K. Then A is free as a left (resp. right) L-module if and only if A is free as a left (resp. right) KL-module.

Proof. We prove the statement for left modules, the proof for right modules is analogous. Note that the commutative L-algebra KL is isomorphic to K ⊗_{F} L, which is free as an L-module. Hence, if A is free as a left KL-module, then it is free as a left L-module.

Suppose conversely that A is free as a left L-module. Then K ⊗_{F} A is free as a left K ⊗_{F} L-module. We have K ⊗_{F} K ≃ K^{[K:F]} as K-modules and thus obtain isomorphisms of left K ⊗_{F} L-modules

K ⊗_{F} A ≃ K ⊗_{F} K ⊗_{K} A ≃ K^{[K:F]} ⊗_{K} A ≃ A^{[K:F]} .

Identifying K ⊗_{F} L with KL we conclude that A^{[K:F]} is free as a KL-module. We will show that this is only possible if A itself is free as a left KL-module.

Since K is étale and F -linearly disjoint from L, it follows from [5, Chap. V, §6, N^{∘} 7] that KL ≃ K_{1} × ⋯ × K_{r} for some fields K_{1}, …, K_{r}. Consider a finitely generated module M over K_{1} × ⋯ × K_{r}. Then M is of the form M_{1} × ⋯ × M_{r} where M_{i} is a K_{i}-vector space for i = 1, …, r. Furthermore M is free if and only if the dimensions dim_{K_{i}} M_{i} for i = 1, …, r are all the same. In particular, M^{n} is free for an arbitrary positive integer n if and only if M is free. □

We call the F -algebra A central simple if dim_{F} A < ∞, Z(A) = F and A is simple as a ring. Let A be a finite-dimensional simple F -algebra. Then K = Z(A) is a field and A is a central simple K-algebra. By Wedderburn’s Theorem (cf. [8, Theorem 2.1.3]) we have dim_{K} A = n^{2} for some positive integer n, which is called the degree of A and denoted by deg A. Moreover, A is Brauer equivalent to a central division K-algebra D, which is is unique up to K-isomorphism. The degree of D is called the index of A and denoted by ind A. If ind A = 1 then A is K-isomorphic to 𝕄_{n}(K) for n = deg A, and in this case we say that A is split. We further set coind A = \frac{deg A}{ind A} and call this the coindex of A. Hence, for any finite-dimensional division F -algebra D and any positive integer n we have coind 𝕄_{n}(D) = n.

Let A be an F -algebra. For any F -subalgebra B of A we obtain an F - subalgebra

C_{A}(B) = {x ∈ A | xb = bx for all b ∈ B},

called the centraliser of B in A.

An element e ∈ A is called an idempotent if e^{2} = e. For any nonzero idempotent e ∈ A the ring eAe with unity e becomes an F -algebra by identifying F with eF. Moreover, if A is a central simple F -algebra, then the F -algebra eAe is also central simple, and it is Brauer equivalent to A.

A crucial tool in the study of central simple algebras and their simple subalgebras is the Double Centraliser Theorem. We refer to [11, Sect. 12.7] for the statement. The following is an extension of this statement for the case of commutative subalgebras.

2.3. Proposition. Let A be a finite-dimensional simple F -algebra. Assume that Z(A) is separable over F. Let L be a commutative semisimple F -subalgebra of A that is F -linearly disjoint from Z(A). Then C_{A}(L) is a semisimple F -algebra with centre L and

[L : F] · dim_{F} C_{A}(L) ⩾ dim_{F} A.

Moreover, the following conditions are equivalent:

(a) [L : F] · dim_{F} C_{A}(L) = dim_{F} A;

(b) all simple components of C_{A}(L) have the same degree;

(c) A is free as a left L-module;

(d) A is free as a right L-module. They hold in particular whenever L is a field or [L : F] = deg A.

Proof. Let K = Z(A). Note that [KL : K] = [L : F], C_{A}(KL) = C_{A}(L) and further that dim_{F} C_{A}(L) = [K : F]·dim_{K} C_{A}(KL) and dim_{F} A = [K : F]·dim_{K} A. In view of the statement and of Lemma 2.2, we may therefore replace K by F and LK by L. Hence we may assume in the sequel that A is central simple as an F -algebra.

If now L is a field, then (a) holds by the Double Centraliser Theorem and furthermore conditions (b)–(d) are trivially satisfied. This case will be used to show the statement in general.

More generally, let e_{1}, …, e_{r} be the primitive idempotents of L. For i = 1, …, r we set A_{i} = e_{i}Ae_{i} and L_{i} = e_{i}L. Thus, identifying F with F e_{i} ⊆ L_{i}, each L_{i} is a finite field extension of F contained in the central simple F -algebra A_{i}, and in the decomposition A = ⨁_{i,j=1}^{r} e_{i}Ae_{j} we have

C_{A}(L) = C_{A_{1}}(L_{1}) ⊕ ⋯ ⊕ C_{A_{r}}(L_{r}).

Each C_{A_{i}}(L_{i}) is a simple F -algebra with centre L_{i}, whereby C_{A}(L) is a semisimple algebra with centre L. For i = 1, …, r we set ℓ_{i} = [L_{i} : F] and d_{i} = deg C_{A_{i}}(L_{i}) and obtain from the Double Centraliser Theorem that

dim_{F} A_{i} = [L_{i} : F] · dim_{F} C_{A_{i}}(L_{i}) = ℓ_{i}^{2}d_{i}^{2},

whereby deg A_{i} = ℓ_{i}d_{i}. It follows that deg A = ∑_{i=1}^{r} ℓ_{i}d_{i} and

dim_{F} C_{A}(L) = ∑_{i=1}^{r} dim_{F} C_{A_{i}}(L_{i}) = ∑_{i=1}^{r} ℓ_{i}d_{i}^{2}.

As [L : F] = ∑_{i=1}^{r} ℓ_{i} it follows that

[L : F] · dim_{F} C_{A}(L) = (∑_{i=1}^{r} ℓ_{i}) · (∑_{i=1}^{r} ℓ_{i}d_{i}^{2})

= (∑_{i=1}^{r} ℓ_{i}d_{i})^{2} + ∑_{i<j} ℓ_{i}ℓ_{j}(d_{i} − d_{j})^{2}
= (deg A)^{2} + ∑_{i<j} ℓ_{i}ℓ_{j}(d_{i} − d_{j})^{2}.

This proves the inequality in the statement as well as the equivalence of (a) and (b) because the last term on the right hand side vanishes if and only if d_{1} = ⋯ = d_{r}. To prove the equivalence of (b) with (c), note that for i = 1, …, r we have

dim_{F} e_{i}A = deg A_{i} · deg A = ℓ_{i}d_{i} deg A,

as one sees easily by reduction to the split case, and thus dim_{L_{i}} e_{i}A = d_{i} deg A. Hence Condition (b) holds if and only if dim_{L_{1}} e_{1}A = ⋯ = dim_{L_{r}} e_{r}A, which is Condition (c). The proof of the equivalence of (b) with (d) is completely analogous.

Finally, if [L : F] = deg A, then ∑_{i=1}^{r} ℓ_{i} = [L : F] = deg A = ∑_{i=1}^{r} ℓ_{i}d_{i}, and thus d_{1} = ⋯ = d_{r} = 1, which implies Condition (b). □

Let K be a quadratic étale F -subalgebra of A and let γ denote its nontrivial F -automorphism. We denote

C′_{A}(K) = {x ∈ A | xk = γ(k)x for all k ∈ K} .

If A is a finite-dimensional semisimple F -algebra then for an element a ∈ A we denote by

Prd_{A,a}(X) ∈ F [X]

its reduced characteristic polynomial (see [12, §9]).

2.4. Proposition. Assume that A is a central simple F -algebra. Let K be an F - subalgebra of A isomorphic to F × F. Let e_{1} and e_{2} be the primitive idempotents of K and A_{i} = e_{i}Ae_{i} for i = 1, 2. Suppose that deg A_{1} = deg A_{2}. Let a ∈ C′_{A}(K), u = e_{1}ae_{2} and v = e_{2}ae_{1}. Then a = u + v and

Prd_{A,a}(X) = Prd_{A_{1},uv}(X^{2}) = Prd_{A_{2},vu}(X^{2}) ∈ F [X] .

Proof. We have that ae_{1} = e_{2}a and ae_{2} = e_{1}a, hence e_{1}a = e_{1}ae_{2} = u and e_{2}a = e_{2}ae_{1} = v, which yields that a = e_{1}a + e_{2}a = u + v,

uv = e_{1}ae_{2}ae_{1} = e_{1}a^{2}e_{1} ∈ A_{1} and vu = e_{2}ae_{1}ae_{2} = e_{2}a^{2}e_{2} ∈ A_{2}.

To prove the equalities we may extend scalars to a splitting field of A. Thus we assume that A is split and identify A with a matrix algebra in such a way that e_{1} = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ), e_{2} = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) and a = ( \begin{matrix}0 & u \\ v & 0 \end{matrix} ), where u and v are matrices. We have to show that the characteristic polynomials Pc_{a}, Pc_{uv}, Pc_{vu} are related by

Pc_{a}(X) = Pc_{uv}(X^{2}) = Pc_{vu}(X^{2}).

Since the coefficients of the characteristic polynomials are polynomial functions of the entries, it suffices to prove these equalities in the case where u and v are generic matrices over ℤ, for the general case then follows by specialization. Since we have Tr(a^{2k}) = 2 Tr((uv)^{k}) = 2 Tr((vu)^{k}) and Tr(a^{2k+1}) = 0 for any k ∈ ℕ, we obtain the result by applying Newton’s Identities relating the coefficients of the characteristic polynomial of a matrix to the traces of its powers. □

2.5. Corollary. Assume that A is a central simple F -algebra. Let K be an étale quadratic F -subalgebra of A such that dim_{F} C_{A}(K) = \frac{1}{2} dim_{F} A. Then

Prd_{A,a}(X) = Prd_{C_{A}(K),a^{2}}(X^{2}) ∈ F [X] for any a ∈ C′_{A}(K) .

Proof. To prove the equality we may extend scalars. Hence we may assume that K ≃ F × F. Then the equation follows from Proposition 2.4. □

3. Capacity

In this section we recall some basic facts and objects associated with involutions on central simple algebras. We recall the distinction of involutions into two kinds and into three different types. We further introduce some notation that will allow us to study involutions of different types and over fields of arbitrary characteristic in a unified way. Our main reference for involutions is [9].

Let A be an F -algebra. By an F -involution on A we mean an F -linear antiautomorphism σ : A → A such that σ ∘ σ = id_{A}. Given an F -involution on A we set

Sym(σ) = {x ∈ A | σ(x) = x} ,
Skew(σ) = {x ∈ A | σ(x) = −x} ,
Symd(σ) = {x + σ(x) | x ∈ A} .

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 with F = Z(A) ∩ Sym(σ) and such that A has no non-trivial two-sided ideal I with σ(I) = I.

In the sequel, let (A, σ) denote an F -algebra with involution. Then either Z(A) = F or Z(A) is a quadratic étale extension of F with non-trivial automorphism σ|_{Z(A)}. One says that (A, σ), or the involution σ, is of the first kind or of the second kind, respectively, according to whether [Z(A) : F] equals 1 or 2.

As long as Z(A) is a field it follows that A is central simple as a Z(A)-algebra. However, if (A, σ) is of the second kind, we may also have that Z(A) ≃ F ×F: in this case (A, σ) ≃ (A_{0} × A_{0}^{op}, sw) for a central simple F -algebra A_{0} and where sw is the so-called switch-involution given by sw(a_{1}, a_{2}^{op}) = (a_{2}, a_{1}^{op}) (see [9, (2.14)]).

If σ is an involution of the first kind, then we say that σ is symplectic if dim_{F} Symd(σ) < dim_{F} Skew(σ) and 1 ∈ Symd(σ), otherwise we say that σ is orthogonal. Considering the F -linear map x ↦ x + σ(x) one sees that

dim_{F} Skew(σ) + dim_{F} Symd(σ) = dim_{F} A,

hence dim_{F} Symd(σ) < dim_{F} Skew(σ) if and only if dim_{F} Symd(σ) < \frac{1}{2} dim_{F} A.

If σ is of the second kind then we also say that σ is unitary. We say that (A, σ) is unitary of inner type when Z(A) ≃ F × F. (The term is motivated by a corresponding notion for algebraic groups.)

The property of the involution σ to be orthogonal, symplectic or unitary is called its type. Notions for properties of an involution (such as its kind and its type) shall also be employed for the algebra with involution as a pair.

Whenever Z(A) is a field we denote by deg A, ind A, coind A the degree, index or coindex of A, respectively, as a central simple Z(A)-algebra. In the case where Z(A) ≃ F × F, we define the same terms with reference to (any of) the two simple components of A. We say that the algebra with involution (A, σ) is split if ind A = 1.

We have Symd(σ) ⊆ Sym(σ), and this is an equality unless char F = 2 and (A, σ) is of the first kind. (See [9, (2.17)] for char F = 2 and σ unitary.) To avoid case distinctions in our statements and arguments, we set

Sym^{∗}(σ) = \begin{cases}Sym(σ) & if σ is orthogonal or unitary, \\ Symd(σ) & if σ is symplectic. \end{cases}

Note that Sym^{∗}(σ) = Symd(σ) except when char F = 2 and σ is orthogonal.

Dealing with orthogonal involutions in characteristic two requires additional care, as one may see in the following statement.

3.1. Proposition. Let e be a nonzero idempotent in Sym(σ) and σ_{e} = σ|_{eAe}. Then (eAe, σ_{e}) is an F -algebra with involution of the same kind as (A, σ). Moreover, (eAe, σ_{e}) is of the same type as (A, σ) except when char F = 2, σ is orthogonal and e ∈ Symd(σ), in which case σ_{e} is symplectic.

Proof. Obviously the F -algebra eAe is stable under the F -involution σ_{e}. Let K = Z(A). We first show that Z(eAe) = Ke. For this we may assume that A is split and identify A with End_{K}(V) for a finitely generated free K-module V. Then eAe is identified with End_{K}(eV). Moreover, the K-submodule eV of V is free; if K is a field this is trivial, and otherwise we have (A, σ) ≃ (A_{0}×A_{0}^{op}, sw) for a central simple F -algebra A_{0} and use that e ∈ Sym(σ) to obtain this conclusion. Hence the centres of End_{K}(V) and End_{K}(eV) consist of the scaling maps with scalars from K. This naturally identifies Z(eAe) with Ke.

Hence (eAe, σ_{e}) is an F -algebra with involution of the same kind as (A, σ). To compare the types of the involutions, we only need to consider the case where (A, σ) is of the first kind.

If a ∈ A is such that a + σ(a) equals 1 or e, then eae + σ(eae) = e. Hence, if 1 ∈ Symd(σ) or e ∈ Symd(σ), then e ∈ Symd(σ_{e}). It remains to consider the dimensions of Symd(σ) and Symd(σ_{e}).

Let f = 1 − e, and let n = deg A, r = deg eAe, and s = deg fAf, so that n = r + s. The decomposition

A = eAe ⊕ (eAf ⊕ fAe) ⊕ fAf

is stable under σ, hence

Symd(σ) = (eAe ∩ Symd(σ)) ⊕ ((eAf ⊕ fAe) ∩ Symd(σ)) ⊕ (fAf ∩ Symd(σ)).

If a ∈ A satisfies a + σ(a) ∈ eAe, then a + σ(a) = eae + σ(eae). This shows that

eAe ∩ Symd(σ) = Symd(σ_{e}),

and it follows that dim_{F} (eAe ∩ Symd(σ)) = \frac{1}{2}r(r + ε_{e}) with ε_{e} = ±1. Likewise, dim_{F} (fAf ∩ Symd(σ)) = \frac{1}{2}s(s + ε_{f}) with ε_{f} = ±1. Now, if a ∈ A is such that a + σ(a) = ebf + fce for some b, c ∈ A, then

ebf + fce = σ(ebf + fce) = eσ(c)f + fσ(b)e,

hence fce = fσ(b)e = σ(ebf) and a + σ(a) = ebf + σ(ebf). Therefore

(eAf ⊕ fAe) ∩ Symd(σ) = {x + σ(x) | x ∈ eAf},

and it follows that dim_{F} (eAf ⊕ fAe) ∩ Symd(σ) = dim_{F} eAf = rs. Therefore the above decomposition of Symd(σ) yields

dim_{F} Symd(σ) = \frac{1}{2}r(r + ε_{e}) + rs + \frac{1}{2}s(s + ε_{f}) = \frac{1}{2}(n^{2} + rε_{e} + sε_{f}).

As dim_{F} Symd(σ) = \frac{1}{2}n(n + ε) for ε = ±1, we conclude that ε = ε_{e} = ε_{f}. Hence, dim_{F} Symd(σ) < \frac{1}{2} dim_{F} A if and only if dim_{F} Symd(σ_{e}) < \frac{1}{2} dim_{F} eAe. □

We give an example for the exceptional case in the statement of Proposition 3.1.

3.2. Example. Write t for the transpose involution on 𝕄_{4}(F). Consider the matrices

m = (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix}) and e = (\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})

in 𝕄_{4}(F). The involution σ = Int(m) ∘ t is orthogonal because m is not alternating. It is further easy to see that e ∈ Symd(σ). Hence, if char(F) = 2, we obtain that σ restricts to a symplectic involution on e𝕄_{4}(F)e.

We define

cap(A, σ) = \{ \begin{matrix}deg A & if σ is orthogonal or unitary, \\ \frac{1}{2} deg A & if σ is symplectic, \end{matrix}

and we call this integer the capacity of (A, σ). This terminology is inspired by the theory of Jordan algebras: when the characteristic is different from 2 and the algebra A is split, then Sym(σ) is a Jordan algebra of capacity equal to cap(A, σ); see [10, § I.5.1]. Note that with this definition there exist F -algebras with involution of any given type and any positive integer as capacity.

3.3. Proposition. Let L be an F -subalgebra of A contained in Sym(σ) and such that L/F is a separable field extension. Set C = C_{A}(L) and σ_{C} = σ|_{C}. Then (C, σ_{C}) is an L-algebra with involution of the same type as (A, σ) and such that

cap(A, σ) = [L : F] · cap(C, σ_{C}) .

Proof. With σ(L) = L we also have that σ(C) = C. Let K = Z(A). Then [LK : L] = [K : F] and [LK : K] = [L : F]. The Double Centraliser Theorem yields that C_{A}(C) = KL and deg A = [L : F] · deg C. As obviously KL ⊆ Z(C) ⊆ C_{A}(C) we conclude that Z(C) = KL. Hence, (C, σ_{C}) is an L-algebra with involution, and σ_{C} is unitary if and only if σ is unitary. Using [9, (4.12)] in the cases where σ is of the first kind, we conclude that the L-algebra with involution (C, σ_{C}) has the same type as the F -algebra with involution (A, σ). Since deg A = [L : F]·deg C, this implies the claimed equality for the capacity. □

We are going to show in Theorem 4.1 that the capacity of (A, σ) is equal to the maximal degree [L : F] where L is an étale F -algebra contained in Sym^{∗}(σ). To this end we first consider the case where A is split and show that we even find then a split étale subalgebra in Sym^{∗}(σ) of degree equal to cap(A, σ).

3.4. Proposition. Let d = cap(A, σ) and assume that (A, σ) is split. Then Sym^{∗}(σ) contains an F -subalgebra L of A with L ≃ F^{d}.

Proof. Assume first that Z(A) is not a field. Then (A, σ) can be identified with (𝕄_{d}(F) × 𝕄_{d}(F)^{op}, sw). Letting L_{0} ⊆ 𝕄_{d}(F) be the algebra of diagonal matrices and L = {(x, x) | x ∈ L_{0}}, we obtain that L ⊆ Sym^{∗}(σ) and L ≃ F^{d}.

Assume now that K = Z(A) is a field. We identify A with End_{K} V for some Kvector space V. Then σ is the adjoint involution of some nondegenerate F -bilinear form b : V × V → K, which is symmetric and non-alternating if σ is orthogonal, which is alternating if σ is symplectic, and which is hermitian with respect to the nontrivial F -automorphism of K if σ is unitary. If σ is orthogonal or unitary (resp. symplectic), we have dim_{K} V = d (resp. dim_{K} V = 2d), and we obtain a decomposition of V into a direct sum of 1-dimensional (resp. 2-dimensional) K-subspaces

V = V_{1} ⊕ … ⊕ V_{d}

that is an orthogonal decomposition for b. Let e_{1}, …, e_{d} denote the orthogonal projections corresponding to this decomposition. Then e_{1}, …, e_{d} are idempotents in A = End_{K} V. For i = 1, …, d and x, y ∈ V we have

b(x, e_{i}(y)) = b(e_{i}(x), e_{i}(y)) = b(e_{i}(x), y).

Thus e_{1}, …, e_{d} ∈ Sym(σ), and we conclude by Proposition 3.1 that e_{1}, …, e_{d} ∈ Sym^{∗}(σ). Hence L = F e_{1} ⊕ ⋯ ⊕ F e_{d} is an F -subalgebra of A contained in Sym^{∗}(σ) with L ≃ F^{d}. □

4. Forms on the space of symmetrized elements

Certain statements on the existence for elements or subalgebras with special properties in an algebra with involution can be proven by reducing to the situation where the base field is algebraically closed. This requires a geometric description of the property in question. Here we are interested in elements and subalgebras contained in Sym^{∗}(σ). To obtain a geometric formulation we introduce a polynomial χ_{a} ∈ F [X] associated to an arbitrary element a ∈ Sym^{∗}(σ), whose degree is equal to cap(A, σ) and which has a as a root. It is defined as either the reduced characteristic polynomial Prd_{A,a} or the Pfaffian characteristic polynomial Prp_{σ,a} (see [9, (2.10)]):

χ_{a} = \begin{cases}Prd_{A,a} & if σ is orthogonal or unitary, \\ Prp_{σ,a} & if σ is symplectic. \end{cases}

For the unitary case, note that, even though the coefficients of the reduced characteristic polynomial of any a ∈ A lie in Z(A), when σ(a) = a the coefficients of Prd_{A,a} lie in F (see [9, (2.16)]). (When (A, σ) = (A_{0} ×A_{0}^{op}, sw), then a = (a_{0}, a_{0}^{op}) for some a_{0} ∈ A_{0}, and χ_{a} = Prd_{A_{0},a_{0}}.) Thus we have χ_{a} ∈ F [X] in all cases. Note that χ_{a} is a multiple of the minimal polynomial of a over F and that the two polynomials have the same irreducible factors. Therefore, if χ_{a} is separable then χ_{a} is the minimal polynomial of a over F.

4.1. Theorem. Any étale F -subalgebra of A contained in Sym(σ) is contained in Sym^{∗}(σ). Furthermore

cap(A, σ) = max{[L : F] | L étale F -algebra with L ⊆ Sym(σ)}.

Proof. Let L ⊆ Sym(σ) be an étale F -algebra. To show that L ⊆ Sym^{∗}(σ) and [L : F] ⩽ cap(A, σ), we may extend scalars and assume that F is algebraically closed. Then L and Z(A) are split. Let r = [L : F] and let e_{1}, …, e_{r} ∈ L be the primitive idempotents in L. Then e_{i}e_{j} = δ_{ij} for i, j ∈ {1, … r} and ∑_{i=1}^{r} e_{i} = 1. It follows that

deg A = ∑_{i=1}^{r} deg(e_{i}Ae_{i}).

If σ is orthogonal or unitary, then we have Sym^{∗}(σ) = Sym(σ) and further cap(A, σ) = deg(A) ⩾ r, because deg(e_{i}Ae_{i}) ⩾ 1 for i = 1, …, r. Assume now that σ is symplectic. Then Proposition 3.1 shows for i = 1, …, r that σ restricts on e_{i}Ae_{i} to a symplectic involution, whereby deg(e_{i}Ae_{i}) ⩾ 2 and e_{i} ∈ Symd(σ). We conclude that L ⊆ Symd(σ) and cap(A, σ) = \frac{1}{2} deg A = ∑_{i=1}^{r} \frac{1}{2} deg e_{i}Ae_{i} ⩾ r. This shows that L ⊆ Sym^{∗}(σ) and [L : F] = r ⩽ cap(A, σ) in any case.

Back in the situation where F is an arbitrary field, it remains to show that Sym^{∗}(σ) contains an étale F -algebra L with [L : F] = cap(A, σ). This follows from Proposition 3.4 if (A, σ) is split. In particular, we may assume that F is infinite. Let \overline{F} denote an algebraic closure of F. Then Sym^{∗}(σ) is Zariski-dense in \overline{Sym^{∗}(σ)} = Sym^{∗}(σ) ⊗_{F} \overline{F}. An element a ∈ \overline{Sym^{∗}(σ)} is separable over \overline{F} if and only if the discriminant of χ_{a} is nonzero. Since this is a polynomial condition, the elements of \overline{Sym^{∗}(σ)} which are separable over \overline{F} form an open subset of \overline{Sym^{∗}(σ)}, and by Proposition 3.4 and Lemma 2.1 this subset is not empty. Since Sym^{∗}(σ) is dense in \overline{Sym^{∗}(σ)}, we conclude that there exists an element a ∈ Sym^{∗}(σ) which is separable over \overline{F}, and thus separable over F. Hence χ_{a} is equal to the minimal polynomial of a over F. We conclude that F [a] is an étale F -algebra and [F [a] : F] = deg(χ_{a}) = cap(A, σ). □

In the context of the last result we observe that Sym^{∗}(σ) may contain (nonétale) commutative F -algebras L with [L : F] > cap(A, σ).

4.2. Example. Let L_{0} be the F -subalgebra of 𝕄_{4}(F) consisting of the matrices

(\begin{matrix}a & 0 & b & c \\ 0 & a & d & e \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & a \end{matrix})

with a, b, c, d, e ∈ F. Then L = {(x, x) | x ∈ L_{0}} is a 5-dimensional commutative F -subalgebra of A = 𝕄_{4}(F) × 𝕄_{4}(F)^{op}. For the involution σ = sw on A we have that cap(A, σ) = 4 and L ⊆ Sym(σ) = Sym^{∗}(σ). 4.3. Proposition. Let Ψ : (A, σ) → (B, τ) be a homomorphism of F -algebras with involution with cap(A, σ) = cap(B, τ). Then χ_{Ψ(a)} = χ_{a} holds for every a ∈ Sym^{∗}(σ). Furthermore, if (A, σ) and (B, τ) are of the same type, then Ψ is an isomorphism. Proof. The characteristic polynomial is invariant under algebra isomorphisms and under scalar extension. Hence, the first part of the statement is obtained by extending scalars to an algebraically closure, where it is easy to verify. Finally, if (A, σ) and (B, τ) are of the same type then dim_{F} A = dim_{F} B and since Ψ is injective, it follows that it is an isomorphism of algebras with involution. □

We give some examples of split algebras with involution and embeddings between them. For a matrix α with coefficients in a ring we denote by α^{t} the transpose matrix of α. If m is a positive integer and α and β are two m × m matrices over a ring, then we denote by α × β the 2m × 2m matrix ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ). 4.4. Proposition. Let m be a positive integer. Let

s : 𝕄_{2m}(F) → 𝕄_{2m}(F), ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ⟼ ( \begin{matrix}δ^{t} & −β^{t} \\ −γ^{t} & α^{t} \end{matrix} ) .

We have the following:

(a) (𝕄_{2m}(F), s) is an F -algebra with symplectic involution of capacity m.

(b) Φ : (𝕄_{m}(F), t) → (𝕄_{2m}(F), s), α ↦ α × α is a homomorphism of F -algebras with involution.

(c) Ψ : (𝕄_{m}(F) × 𝕄_{m}(F)^{op}, sw) → (𝕄_{2m}(F), s), (α, β^{op}) ↦ α × β^{t} is a homomorphism of F -algebras with involution.

Proof. This is obvious. □

We consider the case m = 2. 4.5. Example. Let 𝕀_{2} = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) ∈ 𝕄_{2}(F) and J = ( \begin{matrix}0 & 𝕀_{2} \\ −𝕀_{2} & 0 \end{matrix} ) ∈ 𝕄_{4}(F) . The involution s : 𝕄_{4}(F) → 𝕄_{4}(F) of Proposition 4.4 is given by Int(J) ∘ t, where t is the transposition involution. The F -space Symd(s) consists of the matrices

(\begin{matrix}a & b & 0 & e \\ c & d & −e & 0 \\ 0 & f & a & c \\ −f & 0 & b & d \end{matrix})

with a, b, c, d, e, f ∈ F. For later use we note that the determinant of such a matrix is equal to (ad − bc + ef)^{2}.

Let d = cap(A, σ). For a ∈ Sym^{∗}(σ) we write

χ_{a} = X^{d} − c_{1}(a)X^{d−1} + c_{2}(a)X^{d−2} − ⋯ + (−1)^{d}c_{d}(a)

and observe that this defines a form c_{i}: Sym^{∗}(σ) → F of degree i for i = 1, …, d.

We recall some quadratic form terminology from [6, (7.17)]. Let q : V → F be a quadratic form over F, defined on a finite-dimensional F -vector space V. We denote by b_{q} the polar form of q given by

V × V → F, (x, y) ↦ q(x + y) − q(x) − q(y) .

We further set

rad(b_{q}) = {x ∈ V | b_{q}(x, y) = 0 for all y ∈ V }
rad(q) = {x ∈ rad(b_{q}) | q(x) = 0}

and observe that these are F -subspaces of V with rad(q) ⊆ rad(b_{q}). Moreover, if char F ≠ 2 then q(x) = \frac{1}{2}b_{q}(x, x) for all x ∈ V and thus rad(q) = rad(b_{q}). We call the quadratic form q regular if rad(q) = {0} and nondegenerate if q is regular and dim_{F} rad(b_{q}) ⩽ 1.

4.6. Proposition. Assume that cap(A, σ) = 2 and set V = Sym^{∗}(σ). Then

dim_{F} V = \{ \begin{matrix}3 & if σ is orthogonal, \\ 4 & if σ is unitary, \\ 6 & if σ is symplectic \end{matrix}

and c_{2}|_{V} : V → F is a nondegenerate quadratic form over F, also given by the rule x ↦ x\overline{x} where \overline{x} = c_{1}(x) − x for x ∈ V.

Proof. By the definitions of the capacity and of V the value of dim_{F} V follows from [9, (2.6)]. For x ∈ V we have c_{2}(x) = c_{2}(x) − χ_{x}(x) = −x^{2} + c_{1}(x)x = x\overline{x}.

To show that the quadratic form c_{2} is nondegenerate we may extend scalars and assume that F is algebraically closed. Note that (A, σ) is isomorphic to any F -algebra with involution of same type and of capacity 2. It thus suffices to prove that c_{2} is nondegenerate for a convenient choice of (A, σ).

Consider the F -linear map

Γ : F^{6} → 𝕄_{4}(F) , (a, b, c, d, e, f) ⟼ (\begin{matrix}a & b & 0 & e \\ c & d & −e & 0 \\ 0 & f & a & c \\ −f & 0 & b & d \end{matrix}) .

The characteristic polynomial of Γ(a, b, c, d, e, f) is (T^{2}−(a+d)T +(ad−bc+ef))^{2}, hence its Pfaffian polynomial is T^{2} − (a + d)T + (ad − bc + ef). In particular, c_{2}(Γ(a, b, c, d, e, f)) = ad − bc + ef.

Suppose that σ is symplectic. Then (A, σ) is identified with (𝕄_{4}(F), s), whereby V = Γ(F^{6}). Hence the form c_{2} is given by the polynomial X_{1}X_{4} − X_{2}X_{3} + X_{5}X_{6}, thus it is hyperbolic and in particular nondegenerate.

Suppose that σ is unitary. Then (A, σ) is identified with the image of Ψ, thus V = {Γ(a, b, c, d, 0, 0) | a, b, c, d ∈ F }. Hence, c_{2} is given by the polynomial X_{1}X_{4} − X_{2}X_{3}, thus it is hyperbolic and in particular nondegenerate.

Suppose that σ is orthogonal. Then (A, σ) is identified with the image of Φ. Thus we have V = {Γ(a, b, b, d, 0, 0) | a, b, d ∈ F }. Hence, c_{2} is given by the polynomial X_{1}X_{4} − X_{2}^{2}, thus it is nondegenerate. □

5. Neat subalgebras

Let (A, σ) be an F -algebra with involution. In this section we study étale subalgebras of A that are contained in Sym(σ). An F -subalgebra L of A is called neat in (A, σ) or a neat subalgebra of (A, σ) if L is étale, L ⊆ Sym(σ), A is free as a left L-module and for each nonzero idempotent e of L, the F -algebra with involution (eAe, σ|_{eAe}) has the same type as (A, σ).

5.1. Example. Any separable field extension of F contained in Sym(σ) is neat in (A, σ); this follows by Proposition 3.3, using further Proposition 5.3 below in the case where (A, σ) is unitary of inner type.

By Proposition 3.1, in the definition of neatness the only case where the condition on the idempotents does not follow from the other conditions is when char F = 2 and σ is orthogonal.

5.2. Proposition. Assume that char F = 2 and that σ is orthogonal. Let L be an étale F -subalgebra of A with L ⊆ Sym(σ). Then the following are equivalent:

(i) Symd(σ) contains no nonzero idempotent of L. (ii) Symd(σ) contains no primitive idempotent of L. (iii) For every nonzero idempotent e of L, the involution σ|_{eAe} is orthogonal. In particular, L is neat in (A, σ) if and only if A is free as a left L-module and any of the Conditions (i)–(iii) is satisfied.

Proof. If a nonzero idempotent e of L is contained in Symd(σ), then there exists a primitive idempotent e′ of L with e′e = e′, and writing e = a + σ(a) with a ∈ A we obtain that e′ = e′ae′ + σ(e′ae′) ∈ Symd(σ). Hence (i) and (ii) are equivalent.

For any nonzero idempotent e ∈ L, the involution σ|_{eAe} on eAe is symplectic if and only if e ∈ Symd(σ). This shows that (i) and (iii) are equivalent. □

Neat subalgebras of algebras with unitary involution of inner type are described in the next proposition.

5.3. Proposition. Assume that (A, σ) = (A_{0} × A_{0}^{op}, sw) for a central simple F - algebra A_{0}. Any étale F -subalgebra L_{0} of A_{0} gives rise to an étale F -subalgebra L = {(x, x^{op}) | x ∈ L_{0}} of A contained in Sym(σ) and isomorphic to L_{0}. Conversely, any étale F -subalgebra L of A contained in Sym(σ) is obtained in this way. Furthermore, L is neat in (A, σ) if and only if A_{0} is free as an L_{0}-left module.

Proof. This is obvious. □

The previous proposition allows us to reformulate the condition of neatness in different ways in all cases complementary to the case treated in Proposition 5.2.

5.4. Proposition. Assume that char F ≠ 2 or that σ is symplectic or unitary. Let L be an étale F -subalgebra of A with L ⊆ Sym(σ). The following conditions are equivalent:

(a) [L : F] · dim_{F} C_{A}(L) = dim_{F} A;

(b) all simple components of C_{A}(L) have the same degree;

(c) L is neat in (A, σ).

Proof. In view of the hypothesis and Proposition 3.1, L is neat in (A, σ) as soon as A is free as a left L-module. If A is simple, then the equivalences follow directly from Proposition 2.3. If A is not simple, then (A, σ) ≃ (A_{0}×A_{0}^{op}, sw) for a central simple F -algebra A_{0} and the equivalences follow by Proposition 5.3 along with Proposition 2.3. □

The following proposition shows that the notion of neat subalgebra is preserved under scalar extension.

5.5. Proposition. Let L be a commutative F -subalgebra of (A, σ) and let F′/F be a field extension. Then L ⊗_{F} F′ is neat in (A_{F′}, σ_{F′}) if and only if L is neat in (A, σ).

Proof. Let L′ = L⊗_{F} F′, σ′ = σ_{F′} and A′ = A_{F′}. Clearly the F′-algebra L′ is étale resp. contained in Sym(σ′) if and only if the F -algebra L is étale resp. contained in Sym(σ). Note that C_{A′}(L′) = C_{A}(L) ⊗_{F} F′. Hence, if char F ≠ 2 or if σ is symplectic or unitary, then the statement follows immediately by Proposition 5.4.

We may therefore assume that char F = 2 and that σ is orthogonal. It follows by Proposition 2.3 that A′ is free as a left L′-module if and only if A is free as a left L-module. Hence, it suffices to check the condition on the idempotents. As Symd(σ) ⊆ Symd(σ′) and L ⊆ L′, if Symd(σ′) does not contain any nonzero idempotents of L′, then Symd(σ) does not contain any nonzero idempotents of L. Hence, if L′ is neat in (A′, σ′), then L is neat in (A, σ).

To show the converse implication, we will first reduce the problem to the case where F′/F is a Galois extension. Observe that if F′/F is either a purely transcendental extension or a purely inseparable algebraic extension, then under scalar extension from F to F′ every separable field extension of F remains a field, whereby L ⊗_{F} F′ does not acquire new idempotents. This observation allows us to reduce to the case where F′/F is a separable algebraic extension. Assuming now that L′ is not neat in (A′, σ′), we may replace F′ by its Galois closure over F; by the implication that is already shown, the fact that L′ is not neat in (A′, σ′) will be conserved.

With the assumption that F′/F is a Galois extension, the Galois group acts naturally on A′ and on L′ by fixing A and L, respectively. In view of Proposition 5.2 we may choose a primitive idempotent e′ of L′ with e′ ∈ Symd(σ′). Let e_{1} = e′, e_{2}, …, e_{r} be the different primitive idempotents obtained from e′ via the Galois action. Let e = e_{1} + ⋯ + e_{r}. Note that e is fixed under the Galois action. Since L is the fixed field of the Galois action on L′, we conclude that e is a nonzero idempotent of L. As e′ ∈ Symd(σ′) we have that e_{1}, …, e_{r} ∈ Symd(σ′) and thus e ∈ Symd(σ′) ∩ A = Symd(σ). This shows that L contains a nonzero idempotent in Symd(σ). Hence L is not neat in (A, σ). □

We next show that any étale subalgebra contained in Symd(σ) and of maximal degree under this condition is neat.

5.6. Proposition. Let L be an étale F -subalgebra of A with L ⊆ Sym(σ) and such that [L : F] = cap(A, σ). Then L is neat in (A, σ).

Proof. By Proposition 5.5, in order to show that L is neat we may extend scalars, hence we may assume that L is split. In the case where (A, σ) is unitary of inner type, the statement readily follows from Proposition 5.3 and Proposition 2.3. We may thus assume that A is simple.

Let r = [L : F] and let e_{1}, …, e_{r} be the primitive idempotents of L. Then L = F e_{1} ⊕ ⋯ ⊕ F e_{r}. For i = 1, …, r, set A_{i} = e_{i}Ae_{i}, d_{i} = deg A_{i} and σ_{i} = σ|_{A_{i}}, whereby (A_{i}, σ_{i}) is an F -algebra with involution with Z(A_{i}) = e_{i}Z(A)e_{i} and e_{i} ∈ Sym^{∗}(σ_{i}), according to Theorem 4.1. By the hypothesis, we have that

r = cap(A, σ) = \frac{1}{d} ∑_{i=1}^{r} d_{i}

where d = 2 if σ is symplectic and d = 1 otherwise. In the case where σ is symplectic, we obtain for i = 1, …, r that σ_{i} is symplectic and e_{i} ∈ Symd(σ_{i}), so that σ_{i} restricts to a symplectic involution on A_{i}, whereby d_{i} is even. In any case we conclude from the above equality that d_{1} = ⋯ = d_{r} = d. Hence A is free as a left L-module.

If char F ≠ 2 or if σ is symplectic or unitary, then it follows by Proposition 5.4 that L is neat in (A, σ). Suppose that char F = 2 and that σ is orthogonal. Then d_{1} = ⋯ = d_{r} = d = 1. It follows for i = 1, …, r that σ_{i} is orthogonal and therefore e_{i} ∉ Symd(σ_{i}). Hence e_{1}, …, e_{r} ∉ Symd(σ) and we conclude by Proposition 5.2 that L is neat in (A, σ). □

If char F = 2 and σ is orthogonal then there may exist étale F -subalgebras L of A that are maximal in Sym^{∗}(σ) and with [L : F] < cap(A, σ), as the following example illustrates. In any other case one can actually show that étale F -subalgebras of A contained in Sym^{∗}(σ) and maximal for these properties are of degree equal to cap(A, σ).

5.7. Example. Let char F = 2. We enhance Example 3.2, where in the F -algebra A = 𝕄_{4}(F) we considered two matrices m and e and the orthogonal involution σ = Int(m) ∘ t whose restriction to eAe is symplectic. Set

e_{1} = (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) , e_{2} = (\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) ∈ A.

The F -subalgebra L = F e_{1} ⊕ F e_{2} ⊕ F e of A is split étale and maximal with respect to inclusion among the étale subalgebras of Sym(σ) = Sym^{∗}(σ), and yet we have [L : F] = 3 < 4 = cap(A, σ). The F -subalgebra L is not neat in (A, σ) because the restriction of σ to eAe is symplectic, but also because the conditions of Proposition 2.3 do not hold, since [L : F] does not divide [𝕄_{4}(F) : F]. Moreover, the F -algebra L′ = F e_{1} ⊕ F (e_{2} + e) is split étale and contained in Sym^{∗}(σ), but even though the restrictions of σ to e_{1}Ae_{1} and (e_{2} + e)A(e_{2} + e) are orthogonal, L′ is not neat in (A, σ) because the simple components of its centraliser do not have the same dimension.

Turning back to the situation where (A, σ) is an arbitrary F -algebra with involution, our next goal is to characterize neat subalgebras as subalgebras of symmetric étale algebras of dimension cap(A, σ).

5.8. Lemma. Let L be a neat F -subalgebra of (A, σ) and let K be an F -subalgebra of L. If L is free as a K-module, then K is neat in (A, σ).

Proof. If L is free as a K-module, then using that A is free as a left L-module we obtain that A is free as a left K-module and conclude that K is neat in (A, σ) since all the idempotents in K are in L. □

5.9. Theorem. Let K be a commutative F -subalgebra of (A, σ). Then K is neat in (A, σ) if and only if K ⊆ L ⊆ Sym(σ) for some étale F -subalgebra L of A with [L : F] = cap(A, σ) and such that L is free as a K-module. Moreover, if K and (A, σ) are split, then one can choose L to be split.

Proof. Any étale F -subalgebra L of A with L ⊆ Sym(σ) and [L : F] = cap(A, σ) is neat in (A, σ), by Proposition 5.6, and if K is contained in such an F -algebra L which further is free as a K-module, then it follows by Lemma 5.8 that K is neat in (A, σ).

Assume now that K is neat in (A, σ). Let e_{1}, …, e_{r} be the primitive idempotents of K. For i = 1, …, r, set K_{i} = e_{i}K, A_{i} = e_{i}Ae_{i} and σ_{i} = σ|_{A_{i}}. Since K is neat in (A, σ), we have that (A_{i}, σ_{i}) is an F -algebra with involution of the same type as (A, σ) and with Z(A_{i}) = e_{i}Z(A)e_{i}. Moreover, since A is free as a left K-module, all simple components C_{A_{i}}(K_{i}) of C_{A}(K) have the same degree, by Proposition 5.4 if char F ≠ 2 or σ is symplectic or unitary and otherwise by Proposition 2.3. For all i = 1, …, r the K_{i}-algebras with involution (C_{A_{i}}(K_{i}), σ|_{C_{A_{i}}(K_{i})}) have the same capacity, which we denote by c. Fix i ∈ {1, …, r}. By Theorem 4.1 there exists an étale K_{i}-subalgebra L_{i} of C_{A_{i}}(K_{i}) such that L_{i} ⊆ Sym(σ|_{C_{A_{i}}(K_{i})}) and [L_{i} : K_{i}] = c; moreover, if K and (A, σ) are split, then C_{A_{i}}(K_{i}) is split and K_{i} ≃ F, and we can choose L_{i} to be a split étale F -algebra, by Proposition 3.4. By Proposition 3.3 we have cap(A_{i}, σ_{i}) = c · [K_{i} : F] = [L_{i} : F]. Having this for i = 1, …, r, we obtain that L = L_{1}⊕⋯⊕L_{r} is an étale F -subalgebra of (A, σ) contained in Sym(σ) and such that [L : F] = ∑_{i=1}^{r}[L_{i} : F] = ∑_{i=1}^{r} cap(A_{i}, σ_{i}) = cap(A, σ). As a K-module L is free because the dimensions [L_{i} : K_{i}] are the same for all i. □

5.10. Corollary. Let K be a neat F -subalgebra of (A, σ). Then [K : F] divides cap(A, σ).

Proof. This is obvious from Theorem 5.9. □

The following proposition shows how to construct split étale subalgebras in an F -algebra with involution represented as the endomorphism algebra of a hermitian or skew-hermitian space. We refer to [9, §4] for the terminology and basic facts on hermitian forms.

5.11. Proposition. Let D be a finite-dimensional division F -algebra, V a finitedimensional right D-vector space and A = End_{D} V. Let τ be an F -involution on D for which (D, τ) is an F -algebra with involution. Let h : V × V → D be a hermitian or skew-hermitian form with respect to τ and let σ be the F -involution on A = End_{D} V adjoint to h. Let V_{1}, …, V_{r} be D-subspaces of V such that

V = V_{1} ⊕ ⋯ ⊕ V_{r} .

Let e_{1}, …, e_{r} ∈ A denote the projections corresponding to this decomposition and L = F e_{1} ⊕ ⋯ ⊕ F e_{r}. Then the following hold:

(i) L is a split F -étale subalgebra of A with [L : F] = r. (ii) (A, σ) is an F -algebra with involution of the same kind as (D, τ). (iii) L ⊆ Sym(σ) if and only if the decomposition V = V_{1} ⊕⋯⊕V_{r} is orthogonal

with respect to h. (iv) If L ⊆ Sym(σ), then L is neat in (A, σ) if and only if for i = 1, …, r we

have dim_{D} V_{i} = \frac{1}{r} dim_{D} V and h restricts to a non-alternating form on V_{i} in the case where h is non-alternating.

Proof. Part (i) is clear and Part (ii) is [9, (4.2)]. By the definition of L we have L ⊆ Sym(σ) if and only if

h(x, e_{i}(y)) = h(e_{i}(x), e_{i}(y)) = h(e_{i}(x), y)

holds for all i = 1, …, r and x, y ∈ V, which is if and only if the decomposition of V is orthogonal with respect to h. This shows (iii).

We have

C_{A}(L) = e_{1}Ae_{1} ⊕ ⋯ ⊕ e_{r}Ae_{r} = (End_{D} V_{1}) ⊕ ⋯ ⊕ (End_{D} V_{r})

and dim_{D} V_{i} = deg End_{D} V_{i} for i = 1, …, r. Hence, all simple components of C_{A}(L) have the same degree if and only if dim_{D} V_{i} = \frac{1}{r} dim_{D} V for i = 1, …, r. Assuming that L ⊆ Sym(σ), it is clear that for i = 1, …, r the involution σ restricts to an involution of the same type on e_{i}Ae_{i} except possibly if σ is orthogonal and char F = 2, and in that case the condition holds if and only if the restriction of h on V_{i} is non-alternating. This shows (iv). □

5.12. Corollary. Let r be a positive integer. There exists a split neat F -subalgebra of (A, σ) of degree r if and only if r divides coind A and cap(A, σ).

Proof. If (A, σ) is split symplectic, then coind A = 2 cap(A, σ), and in this case we set d = cap(A, σ). In any other case coind A divides cap(A, σ), and we set d = coind A. In view of Corollary 5.10 the degree of any neat F -subalgebra of (A, σ) divides cap(A, σ). On the other hand, the degree of a split neat F - subalgebra of (A, σ) clearly divides d.

We claim that (A, σ) contains a split neat F -subalgebra L of degree d. Once this is shown, assuming that r divides d, we may choose an F -subalgebra K of L with [K : F] = r and such that L is free as a K-module, and obtain that K is split and neat in (A, σ), by Lemma 5.8.

A central simple algebra A_{0} contains an F -subalgebra L_{0} isomorphic to F^{r} and such that A_{0} is free as a left L_{0}-module if and only if r divides coind A_{0}. This together with Proposition 5.3 shows the statement in the case where (A, σ) is unitary of inner type. Hence we may for the rest of the proof assume that A is simple.

Assume that (A, σ) is split. Then d = cap(A, σ) and it follows from Proposition 3.4 together with Proposition 5.6 that (A, σ) contains a split neat F - subalgebra L with [L : F] = cap(A, σ).

For the rest of the proof we may in particular assume that (A, σ) is not split symplectic, whereby d = coind A. We identify A with End_{D} V where D is a non-commutative division F -algebra and V is a finite-dimensional right D-vector space. Then d = dim_{D} V.

Since (A, σ) is not split symplectic we may fix an F -involution τ on D of the same type as σ. Then σ is adjoint to a hermitian form h : V × V → D with respect to τ. Since (D, τ) is in particular not split symplectic, we may diagonalise h. In other words, we find a D-basis (v_{1}, …, v_{d}) of V which is orthogonal for h. Letting a_{i} = h(v_{i}, v_{i}) for i = 1, …, d, we obtain that a_{1}, …, a_{d} ∈ Sym(D, τ) and

h ≃ ⟨a_{1}, …, a_{d}⟩ .

Note that h is alternating if and only if char F = 2 and σ is symplectic, and in this case a_{1}, …, a_{d} ∈ Symd(D, τ). If char F = 2 and σ is orthogonal, then at least one of a_{1}, …, a_{d} is not contained in Symd(D, τ). However, if char F = 2 then for any x ∈ Sym(D, τ) \ Symd(D, τ) and a ∈ Symd(D) \ {0} we have

⟨x, a⟩ ≃ ⟨x + a, x^{−1} + a^{−1}⟩

and x + a, x^{−1} + a^{−1} ∈ Sym(D, τ) \ Symd(D, τ). Hence, if char F = 2 and σ is orthogonal, then one can change the diagonalisation appropriately and assume that a_{1}, …, a_{d} ∈ Sym(D, τ) \ Symd(D, τ). Then the orthogonal basis (v_{1}, …, v_{d}) yields an orthogonal decomposition of (V, h) in r subspaces of equal dimension, and the restriction of h to any of these subspaces is only alternating in the case where char F = 2 and σ is symplectic. By Proposition 5.11 the projections corresponding to this decomposition generate a split neat F -subalgebra of (A, σ) of degree d. □

6. Neat quadratic subalgebras

Throughout this section let (A, σ) be an F -algebra with involution and let K be a neat quadratic F -subalgebra of (A, σ). We shall prove that there exists a maximal neat subalgebra of (A, σ) of the form KL for a neat F -subalgebra L of (A, σ) which is F -linearly disjoint from K. This result will be crucial for our main results in the final section.

Set C = C_{A}(K) and

C′ = C′_{A}(K) = {x ∈ A | xk = γ(k)x for all k ∈ K}

where γ denotes the nontrivial F -automorphism of K.

6.1. Proposition. We have A = C ⊕ C′ and the F -vector spaces C and C′ are stable under σ and satisfy

C ∩ Symd(σ) = Symd(σ|_{C}) and C′ ∩ Sym(σ) = C′ ∩ Symd(σ) .

Moreover, dim_{F} C′ = dim_{F} C = \frac{1}{2} dim_{F} A and dim_{F} (C′ ∩ Sym(σ)) = \frac{1}{4} dim_{F} A.

Proof. We fix u ∈ K \ F with u^{2} − u ∈ F and set c = u^{2} − u. Hence γ(u) = 1 − u and we have 4c + 1 ≠ 0 because the roots of the polynomial X^{2} − X − c are simple. We obtain that C′ = {x ∈ A | xu + ux = x}. Since σ(u) = u we have σ(C′) = C′.

Consider the F -linear map

φ: A → A, x ↦ \frac{1}{4c+1}((2c + 1)x − ux − xu + 2uxu).

Computation shows that φ(x) ∈ C and x − φ(x) ∈ C′ for any x ∈ A, and moreover φ(x) = x for x ∈ C and φ(x) = 0 for x ∈ C′. Therefore

A = C ⊕ C′ .

As K is neat in (A, σ) and [K : F] = 2, it follows from Proposition 2.3 in the case where A is simple and otherwise from Proposition 5.3 that dim_{F} C = \frac{1}{2} dim_{F} A, whereby dim_{F} C′ = \frac{1}{2} dim_{F} A.

Now, consider the F -linear map f: C′ → C′, x ↦ x+σ(x) and set W = ker(f) and U = f(C′). Hence W = C′ ∩ Skew(σ) and dim_{F} C′ = dim_{F} W + dim_{F} U. For x ∈ C′ ∩ Sym(σ) we have x = xu + ux = xu + σ(xu) ∈ U ⊆ C′ ∩ Symd(σ), whereby

U = C′ ∩ Sym(σ) = C′ ∩ Symd(σ) .

As (1 − 2u)^{2} = 1 + 4c ∈ F^{×}, multiplication from the left by 1 − 2u yields F -iso-morphisms between U and W. Hence dim_{F} U = dim_{F} W = \frac{1}{2} dim_{F} C′.

Clearly we have Symd(σ|_{C}) ⊆ C ∩ Symd(σ). To show the converse inclusion, we may obviously assume that char F = 2, whereby u^{2} + u = c ∈ F. Consider z ∈ C ∩ Symd(σ). Let x ∈ A be such that z = σ(x) + x. As u ∈ K ⊆ C we have zu = uz and obtain that σ(xu + ux) = (z + x)u + u(z + x) = ux + xu and thus

z = σ(xu + ux + x) + xu + ux + x .

Since u(ux + xu + x) = (u^{2} + u)x + uxu = x(u^{2} + u) + uxu = (ux + xu + x)u we further have ux + xu + x ∈ C_{A}(u) = C and conclude that z ∈ Symd(σ|_{C}). □

6.2. Remark. The definition of φ in the proof comes from the observation that \frac{1}{4c+1}(2c + 1 − u ⊗ 1 − 1 ⊗ u + 2u ⊗ u) is the separability idempotent of K.

Let us consider in more detail the case where K is a field. We then consider C as a K-algebra and denote by σ_{C} the K-involution on C obtained by restricting σ.

6.3. Proposition. Assume that K is a field. For a ∈ C′ ∩ Sym(σ) we have

a^{2} ∈ Sym^{∗}(σ_{C}) and χ_{A,a}(X) = χ_{C,a^{2}}(X^{2}) ∈ F [X^{2}] .

Proof. As a ∈ C′ ∩ Sym(σ) we have that a^{2} ∈ C ∩ Sym(σ). If σ is symplectic then σ_{C} is symplectic by Proposition 3.3 and for ℓ ∈ C satisfying ℓ + σ(ℓ) = 1 we obtain that a^{2} = aℓa + σ(aℓa). This shows that a^{2} ∈ Sym^{∗}(σ_{C}).

¿From Corollary 2.5 we obtain the equality

Prd_{A,a}(X) = Prd_{C,a^{2}}(X^{2}) .

We conclude that χ_{C,a^{2}}(X^{2}) = χ_{A,a}(X) ∈ F [X^{2}]. □

Back in the more general situation where K is a neat quadratic F -algebra, but not necessarily a field, we conclude the following.

6.4. Corollary. For a ∈ C′ ∩ Sym(σ) we have χ_{a}(X) ∈ F [X^{2}].

Proof. If K ≃ F × F then the statement follows from Proposition 2.4. Otherwise K is a field, so that the statement follows from Proposition 6.3. □

6.5. Proposition. Assume that cap(A, σ) = 2. Then

Sym^{∗}(σ) = K ⊕ (C′ ∩ Sym(σ))

and this decomposition is orthogonal for the quadratic form c_{2} : Sym^{∗}(σ) → F. Furthermore c_{2}|_{K} : K → F is the norm form of K and c_{2}(x) = −x^{2} ∈ F for all x ∈ C′ ∩ Sym(σ).

Proof. Set V = Sym^{∗}(σ) and W = C′ ∩ Sym(σ). It follows from Proposition 6.1 and by comparing dimensions that V = K ⊕ W. Writing \overline{x} = c_{1}(x) − x for x ∈ V defines an F -linear map V → V, x ↦ \overline{x}. By Proposition 4.6 we have c_{2}(x) = x\overline{x} for any x ∈ V.

For x ∈ F we have χ_{x} = (X − x)^{2}, whereby c_{1}(x) = 2x and c_{2}(x) = x^{2}. For x ∈ V \ F we have that χ_{x} is the minimal polynomial of x over F. ¿From this we conclude that \overline{x} = γ(x) for x ∈ K and that c_{2}|_{K} is the norm form of K.

For w ∈ W we have w^{2} ∈ C ∩ Sym^{∗}(σ) ∩ C_{A}(w) = F, whence χ_{w} = X^{2} − w^{2} and w + \overline{w} = c_{1}(w) = 0. For v ∈ K and w ∈ W, using that vw = wγ(v) = w\overline{v} and w + \overline{w} = 0 we obtain that

c_{2}(v + w) − c_{2}(v) − c_{2}(w) = v\overline{w} + w\overline{v} = v(w + \overline{w}) = 0 .

This shows that K is orthogonal to W with respect to c_{2} and that \overline{x} = −x and c_{2}(x) = −x^{2} for x ∈ W. □

6.6. Corollary. If cap(A, σ) = 2, then K is contained in a σ-stable quaternion F -subalgebra of A.

Proof. Let W = C′ ∩Sym(σ). It follows from Proposition 4.6 and Proposition 6.5 that the quadratic form c_{2}|_{W} : W → F, x ↦ −x^{2} is nondegenerate. As W ≠ {0} it follows that there exists x ∈ W with x^{2} ∈ F^{×}. Since Kx ⊆ C′ we conclude that K ⊕ Kx is a σ-stable quaternion F -subalgebra. □

For a ∈ C′ ∩ Sym(σ), by Corollary 6.4 there is a unique polynomial f ∈ F [X] with χ_{a}(X) = f(X^{2}), and we call the element a square separable if f is separable.

6.7. Proposition. Assume that F is algebraically closed. In A the set

{a ∈ C′ ∩ Sym(σ) | a is square separable} ∩ A^{×}

is open in C′ ∩ Sym(σ) with respect to the Zariski topology.

Proof. A polynomial in F [X] is inseparable if and only if its discriminant vanishes. Hence, for a ∈ C′ ∩ Sym(σ) being square separable is characterised by the nonvanishing of a polynomial in the coefficients of χ_{a}(X), which in turn are polynomials in the coefficients of a with respect to any fixed F -basis of C′ ∩ Sym(σ). Therefore in C′ ∩Sym(σ) the square separable elements form an open subset with respect to the Zariski topology. On the other hand, in A the invertible elements are characterised by the nonvanishing of the reduced norm, whereby A^{×} is open in A. The statement follows from these two observations by basic topology. □

6.8. Proposition. If |F | > cap(A, σ) then C′ ∩ Sym(σ) ∩ A^{×} contains a square separable element.

Proof. Suppose first that F is finite or algebraically closed. Then (A, σ) is split. Set r = \frac{1}{2} cap(A, σ). By the hypothesis we have that |F^{×2}| ⩾ r. If K is a field then (C, σ_{C}) is a split K-algebra with involution of the same type as (A, σ) and with cap(C, σ_{C}) = r, so that by Corollary 5.12 there exists a split neat Ksubalgebra L of (C, σ_{C}) with [L : K] = r. If K ≃ F × F then by Theorem 5.9, K is contained in a split neat F -subalgebra L of (A, σ) with [L : F] = 2r. In either of these two cases we have that L ≃ K^{r} as F -algebras. Let e_{1}, …, e_{r} ∈ L be the corresponding idempotents in L satisfying K ≃ Ke_{i} for i = 1, …, r.

If K ≃ F × F, then we fix a primitive idempotent f ∈ K and obtain that e_{1}f, …, e_{r}f, e_{1}(1 − f), …, e_{r}(1 − f) are the primitive idempotents in L. For i = 1, …, r, we set A_{i} = e_{i}Ae_{i} and σ_{i} = σ|_{A_{i}} and obtain by identifying F with F e_{i} ⊆ A_{i} that (A_{i}, σ_{i}) is a split F -algebra with involution of the same type as (A, σ) and with cap(A_{i}, σ_{i}) = \frac{1}{r} cap(A, σ) = 2.

Consider i ∈ {1, …, r}. By Corollary 6.6 the neat quadratic F -subalgebra Ke_{i} of A_{i} is contained in a σ_{i}-stable quaternion F -subalgebra Q_{i} of A_{i}. By the assumption on F, Q_{i} is split, and since Ke_{i} ⊆ Sym(σ_{i}) we obtain by Theorem 4.1 that σ_{i}|_{Q_{i}} is orthogonal. It follows that there exists g_{i} ∈ Q_{i} ∩ Sym(σ_{i}) with g_{i}^{2} ∈ F e_{i} and such that Int_{Q_{i}}(g_{i}) restricts to the nontrivial F -automorphism on Ke_{i}. Note that g_{i} is determined by this property up to a multiple in F^{×}. Moreover, since |F^{×2}| ⩾ r, we may choose g_{1}, …, g_{r} in such way that g_{i}^{2} = c_{i}e_{i} for i ∈ {1, …, r} where c_{1}, …, c_{r} ∈ F^{×} are pairwise distinct. For g = g_{1} + ⋯ + g_{r} ∈ C′ ∩ Sym(σ) it follows that Prd_{A,g} = ∏_{i=1}^{r}(X^{2} − c_{i}), whereby g is invertible and square separable.

We turn to the general case, where we may assume that F is an infinite field. We choose an algebraic closure \overline{F} of F and consider the \overline{F} -algebra with involution (\overline{A}, \overline{σ}) naturally obtained from (A, σ) by letting \overline{A} = A ⊗_{F} \overline{F} and \overline{σ} = σ ⊗ id\overline{F}. For any F -subspace W of A we write \overline{W} = W ⊗_{F} \overline{F} and note that W is dense in \overline{W} for the Zariski topology. By the above and by Proposition 6.7 the elements of \overline{C′} ∩ Sym(\overline{σ}) that are square separable and invertible in \overline{A} form a nonempty Zariski-open subset of \overline{C′} ∩ Sym(\overline{σ}). As \overline{C′} ∩ Sym(\overline{σ}) = \overline{C′ ∩ Sym(σ)}, we obtain that there exists a square separable invertible element in C′ ∩ Sym(σ). □ 6.9. Proposition. Let a ∈ C′ ∩ Sym(σ) ∩ A^{×} be square separable. Then F [a^{2}] and K[a^{2}] are neat F -subalgebras of (A, σ) such that F [a^{2}] is F -linearly disjoint from K and [K[a^{2}] : F] = [F [a] : F] = cap(A, σ). Proof. By the hypothesis χ_{a}(X) = f(X^{2}) for a separable polynomial f ∈ F [X]. In particular deg f = \frac{1}{2} deg χ_{a} = \frac{1}{2} cap(A, σ). As f is separable and f(a^{2}) = 0, the F -algebra F [a^{2}] is étale and [F [a^{2}] : F] = deg f = \frac{1}{2} cap(A, σ). Since a ∈ C′ we have that a^{2} ∈ C, whereby F [a^{2}] ⊆ C and aF [a^{2}] ⊆ C′, which shows that F [a] = F [a^{2}] ⊕ aF [a^{2}]. Since a ∈ A^{×} it follows that

[F [a] : F] = 2 · [F [a^{2}] : F] = cap(A, σ) .

As a^{2} ∈ C the F -algebra K[a^{2}] is commutative. Conjugation by a restricts to a nontrivial F -automorphism of order two on K[a^{2}] which fixes F [a^{2}]. It follows that K is F -linearly disjoint from F [a^{2}] and that [K[a^{2}] : F [a^{2}]] = 2. Hence, [K[a^{2}] : F] = cap(A, σ) and K[a^{2}] is neat in (A, σ), by Proposition 5.6. Since K[a^{2}] is free as an F [a^{2}]-module, it follows by Lemma 5.8 that F [a^{2}] is neat in (A, σ). □ 6.10. Theorem. There exists a neat subalgebra L of (A, σ) contained in C_{A}(K), F -linearly disjoint from K and such that KL is a neat subalgebra of (A, σ) with [KL : F] = cap(A, σ).

Proof. Assume that (A, σ) is split. If K ≃ F × F, then the statement follows immediately from Theorem 5.9. Suppose now that K is a field. Set C = C_{A}(K) and σ_{C} = σ|_{C}. By Proposition 3.3 we obtain that (C, σ_{C}) is a split K-algebra with involution such that cap(C, σ_{C}) = \frac{1}{2} cap(A, σ). It follows by Proposition 3.4 that Sym^{∗}(σ_{C}) contains a split étale K-algebra M of A with [M : K] = cap(C, σ_{C}). By Proposition 5.6 we have that M is neat in (A, σ). Let L be the F -subalgebra of M generated by the idempotent elements in M. Then L is F -linearly disjoint from K and KL = M, thus [KL : F] = [M : K] · [K : F] = cap(A, σ). Furthermore, M is free as an L-module, whence L is neat in (A, σ) by Lemma 5.8.

Hence the statement holds when A is split. In particular it holds when F is finite. Assume now that F is infinite. By Proposition 6.8 there exists an element a ∈ C′ ∩ Sym(σ) ∩ A^{×} which is square separable. Then Proposition 6.9 shows that L = F [a^{2}] has the desired property. □

6.11. Remark. If char F ≠ 2 then instead of the set in Proposition 6.7 one may consider the set {a ∈ C′ ∩ Sym(σ) ∩ A^{×} | χ_{a} separable}. To see that this set is Zariski-open in C′ ∩Sym(σ) when F is algebraically closed is easier, as it does not involve Proposition 2.4, Corollary 2.5 and Proposition 6.3. Note however that this set is empty if char F = 2.

7. Capacity four

In this section we consider in more detail algebras with involution of capacity four and show the existence of biquadratic neat subalgebras (Theorem 7.4). We shall in particular be interested in the case of symplectic involutions on algebras of degree eight. In this case we will conclude the existence of a triquadratic étale extension of the centre which is stable under the involution (Theorem 7.6). In particular, we obtain a new proof to Rowen’s Theorem stating that every degree eight algebra of exponent two contains a triquadratic étale subalgebra (Corollary 7.7).

We need the following two preparatory results, which are well-known.

7.1. Proposition. Assume that A is a central simple F -algebra. There exist r ∈ ℕ and a sequence of separable quadratic field extensions (F_{i}/F_{i−1})_{i=1}^{r} with F_{0} = F such that ind A_{F_{r}} is odd.

Proof. Primary decomposition (cf. [8, Proposition 4.5.16]) yields that A ≃ B ⊗ C for two central simple F -algebras B and C such that ind B is odd and ind C = 2^{m} for some m ⩾ 1. Then C represents an element of order dividing 2^{m} in the Brauer group of F. By [2, Theorem] there exist r ∈ ℕ and a sequence of separable quadratic field extensions (F_{i}/F_{i−1})_{i=1}^{r} with F_{0} = F and such that C_{F_{r}} is split. (Alternatively, this can be derived from Merkurjev’s Theorem [8, Theorem 1.5.8].) It follows that ind A_{F_{r}} divides ind B. □

7.2. Proposition (Springer). Any cubic form over F which has a nontrivial zero over a quadratic field extension of F also has a nontrivial zero over F.

Proof. Consider a cubic form f in n variables over F. We suppose that f has a nontrivial zero in F [X]/(p) for some irreducible quadratic polynomial p ∈ F [X]. Hence there exist b_{1}, c_{1}, …, b_{n}, c_{n} ∈ F, not all zero, and h ∈ F [X] such that

f(b_{1} + c_{1}X, …, b_{n} + c_{n}X) = p(X) · h(X) .

Suppose first that h ∈ F. Comparing coefficients in degree 3 we obtain that f(c_{1}, …, c_{n}) = 0. Moreover, if c_{1} = ⋯ = c_{n} = 0 then we obtain further that f(b_{1}, …, b_{n}) = 0. As b_{1}, c_{1}, …, b_{n}, c_{n} ∈ F are not all zero, it follows that f has a nontrivial zero in F.

Suppose now that h ∉ F. As deg(f(b_{1} + c_{1}X, …, b_{n} + c_{n}X)) ⩽ deg(f) = 3 and deg(p) = [K : F] = 2, we conclude that deg(h) = 1. Hence there exists a ∈ F such that h(a) = 0. Then f(b_{1} + c_{1}a, …, b_{n} + c_{n}a) = 0. Moreover, if b_{i} + c_{i}a = 0 for i = 1, …, n, then 0 = f(c_{1}(X − a), …, c_{n}(X − a)) = (X − a)^{3}f(c_{1}, …, c_{n}) and thus f(c_{1}, …, c_{n}) = 0. Hence f has a nontrivial zero in F. □

Now let (A, σ) be an F -algebra with involution. Recall that for d = cap(A, σ) and a ∈ Sym^{∗}(σ) we have

χ_{a} = X^{d} − c_{1}(a)X^{d−1} + c_{2}(a)X^{d−2} + ⋯ + (−1)^{d}c_{d}(a) ∈ F [X] . 7.3. Lemma. Assume that cap(A, σ) is a multiple of 4. There exists an element a ∈ Sym^{∗}(σ) \ F such that c_{1}(a) = c_{3}(a) = 0. Proof. Let d = cap(A, σ). We first consider the situation where coind(A) is even. Then by Corollary 5.12 there exists a split neat F -subalgebra L of (A, σ) with [L : F] = 2. If char F ≠ 2 we choose an element a ∈ L\F with a^{2} = 1 and obtain that χ_{a}(X) = (X^{2} − 1)^{d/2}. If char F = 2 then we choose a ∈ L \ F with a^{2} = a and obtain that χ_{a}(X) = X^{d} +X^{d/2}. In either case we have that a ∈ Sym^{∗}(σ)\F and χ_{a}(X) ∈ F [X^{2}]. Hence a has the desired properties.

In the general case, by Proposition 7.1 there exists r ∈ ℕ and a sequence of quadratic field extensions (F_{i}/F_{i−1})_{i=1}^{r} with F_{0} = F such that ind A_{F_{r}} is odd. In particular, coind A_{F_{r}} is even.

Let W = ker(c_{1}) ⊆ Sym^{∗}(σ). If char F ≠ 2, then F ∩ W = 0, and we consider the cubic form f = c_{3} on W. If char F = 2, then we have c_{3}(x + a) = c_{3}(x) for every x ∈ W and every a ∈ F, for χ_{x+a}(X) = χ_{x}(X −a). In this case we consider the cubic form f: W/F → F, x + F ↦ c_{3}(x). In each case the validity of the statement is equivalent to the existence of a nontrivial zero of the cubic form f. By the special case considered above, f has a nontrivial zero in F_{r}. Since f is a cubic form and F_{i}/F_{i−1} is a quadratic extension for i = 1, …, r, we conclude by Proposition 7.2 that f has a nontrivial zero over F. □

An étale F -algebra K is called biquadratic (resp. triquadratic) if it is isomorphic to the tensor product of two (resp. three) quadratic étale F -algebras.

The following result extends [7, Corollary 6.2 and Theorem 9.1 (1)]. 7.4. Theorem. Let (A, σ) be an F -algebra with involution with cap(A, σ) = 4. Then (A, σ) contains a neat biquadratic F -subalgebra.

Proof. By Theorem 6.10 it suffices to show that (A, σ) contains a neat quadratic F -subalgebra K. If A has zero-divisors, then we may conclude this by applying Corollary 5.12 with r = 2. Hence we assume that A is a division F -algebra.

By Lemma 7.3 there exists an element a ∈ Sym^{∗}(σ)\F with c_{1}(a) = c_{3}(a) = 0, whereby χ_{a}(X) = X^{4} + c_{2}(a)X^{2} + c_{4}(a) ∈ F [X^{2}]. In particular [F (a^{2}) : F] ⩽ 2. We set E = F [a] if a^{2} ∈ F and E = F [a^{2}] otherwise. Then E is a quadratic field extension of F contained in Sym(σ).

If E is separable over F, then E is a neat subalgebra of (A, σ) and we may take K = E. Suppose now that the quadratic extension E is inseparable. In particular char F = 2. We consider C = C_{A}(E) and write σ_{C} for the restriction of σ to C. If we can find y ∈ Sym(σ_{C}) \ E such that y^{2} + y ∈ E, we obtain for u = y^{2} that u ∈ Sym(σ_{C}) \ E ⊆ Sym(σ) \ F and u^{2} + u = (y^{2} + y)^{2} ∈ F, whereby F [u] is a separable quadratic extension of F contained in Sym(σ), so that we may take K = F [u]. It therefore suffices to show the existence of such an element y.

Note that (C, σ_{C}) is an E-algebra with involution and deg(C) = \frac{1}{2} deg(A). If cap(C, σ_{C}) = 2, then the existence of y ∈ Sym(σ_{C}) \ E with y^{2} + y ∈ E follows by Theorem 4.1. The only possibility to have cap(C, σ_{C}) ≠ 2 is that (A, σ) is symplectic of degree 8 and (C, σ_{C}) is orthogonal of degree 4.

In particular, the statement holds in the case where σ is orthogonal. Applying this to (C, σ_{C}) when σ_{C} is orthogonal and deg(C) = 4, we obtain a separable quadratic extension of E inside Sym(σ_{C}) and thus an element y ∈ Sym(σ_{C}) \ E with y^{2} + y ∈ E, as desired. □

The proof of our next result uses a corestriction argument on central simple algebras. Consider a separable quadratic field extension K/F and a central simple K-algebra B. We refer to [9, §3.B] for the definition and the basic properties of the central simple F -algebra Cor_{K/F} (B), the corestriction (or norm) of B from K to F (which is denoted N_{K/F} (B) in [9]).

7.5. Lemma. Assume that B is a K-quaternion algebra. Then B contains a quadratic étale F -algebra linearly disjoint from K if and only if Cor_{K/F} (B) is not a division algebra.

Proof. If char F ≠ 2, a proof is given in [9, (16.28)]. We refer to [3] for a proof in arbitrary characteristic. □

The following result extends [7, Lemma 6.1].

7.6. Theorem. Assume that (A, σ) is symplectic of degree 8. Then A contains a σ-stable triquadratic étale F -subalgebra. Moreover, any neat biquadratic F -sub-algebra of (A, σ) is contained in a σ-stable triquadratic étale F -subalgebra of A.

Proof. In view of Theorem 7.4 it suffices to prove the second part of the statement. Thus let L be a neat biquadratic F -subalgebra of (A, σ).

Assume first that L is split. Let e_{1}, …, e_{4} be the primitive idempotents of L. For i = 1, …, 4 by identifying F with F e_{i} we obtain that e_{i}Ae_{i} is a quaternion F - algebra Brauer equivalent to A and σ|_{e_{i}Ae_{i}} is its canonical involution. As deg A = 8 and A contains a split biquadratic étale F -subalgebra, we have ind A ⩽ 2. If ind A = 2 then we fix a ∈ F with 4a ≠ −1 such that F [X]/(X^{2} − X − a) is a splitting field of A, otherwise we set a = 0. In either case, we obtain for i = 1, …, 4 an element f_{i} ∈ e_{i}Ae_{i}\F e_{i} with f_{i}^{2} = f_{i}+ae_{i}. Then f = f_{1}+f_{2}+f_{3}+f_{4} is such that f + σ(f) = 1 and f^{2} = f + a. Hence L[f] is a σ-stable triquadratic F -subalgebra of A.

Assume now that L is not split. Then L contains a quadratic field extension K of F. With the notation of Section 6 we obtain an F -algebra with involution (C, σ_{C}). By Proposition 3.3 the involution σ_{C} is symplectic and cap(C, σ_{C}) = 2. Since [L : K] = 2 = cap(C, σ_{C}), by Proposition 5.6 the étale K-algebra L is neat in (C, σ_{C}). By Corollary 6.6 it follows that L is contained in a σ_{C}-stable quaternion K-subalgebra Q of C. We set Q′ = C_{C}(Q) and observe that Q′ is a σ_{C}-stable quaternion K-subalgebra of C. We set σ_{Q} = σ|_{Q} and σ_{Q′} = σ|_{Q′} and obtain that (Q, σ_{Q}) and (Q′, σ_{Q′}) are quaternion K-algebras with involution such that

(C, σ_{C}) ≃ (Q, σ_{Q}) ⊗ (Q′, σ_{Q′}) .

Since L ⊆ Sym(σ_{Q}) it follows from Theorem 4.1 that σ_{Q} is orthogonal. As σ_{C} is symplectic, it follows by [9, (2.23)] that σ_{Q′} is symplectic. Hence σ_{Q′} is the canonical involution of Q′.

The central simple K-algebra C is Brauer equivalent to A_{K}. Since A carries an F -linear involution, A ⊗_{F} A is split. This implies that Cor_{K/F} (C) is split, hence Cor_{K/F} (Q) ≃ Cor_{K/F} (Q′). Since L is biquadratic, we have L ≃ K ⊗_{F} M for some quadratic étale F -algebra M. As M ⊆ Q, Lemma 7.5 shows that Cor_{K/F} (Q) is not a division algebra. Therefore Cor_{K/F} (Q′) is not a division algebra. Hence, by Lemma 7.5 there exists a quadratic étale F -algebra K′ ⊆ Q′ linearly disjoint from K. Note that K′ is σ-stable, for σ|_{Q′} is the canonical involution of Q′. Note further that K′ ⊆ Q′ = C_{C}(Q) ⊆ C_{A}(L). Hence LK′ is a σ-stable triquadratic étale F -subalgebra of A. □

It is known that every central division algebra of exponent two and degree at most eight has a maximal subfield that is a separable multiquadratic extension of the centre. This was shown by Albert [1, Chapter XI, Theorem 9] for degree four and by Rowen [13, Theorem 1] for degree eight. We obtain a new proof of this statement.

7.7. Corollary (Albert, Rowen). Let A be a central simple F -algebra such that deg(A) divides 8 and A ⊗_{F} A is split. Then A contains a maximal commutative subalgebra that is an étale multiquadratic F -algebra.

Proof. Let n ∈ ℕ be such that deg(A) = 2^{n}. If n ⩽ 1 then the statement is obvious. If n = 2 then we choose an orthogonal involution σ on A and conclude by Theorem 7.4. If n = 3 then we choose a symplectic involution σ on A and apply Theorem 7.6. □

Acknowledgments. It is a pleasure to thank Holger Petersson for his comments on Jordan algebraic aspects, Kader Bingöl for a critical reading of a preliminary version and the referee for pointing out a gap (and suggesting a patch) in a first version of Corollary 5.12.

This work was supported by the FWO Odysseus Programme (project Explicit Methods in Quadratic Form Theory), funded by the Fonds Wetenschappelijk Onderzoek – Vlaanderen. The third author acknowledges support from the Fonds de la Recherche Scientifique–FNRS under grants n^{∘} J.0014.15 and J.0149.17. Work on this paper was initiated in 2011 while the first and the third author where, respectively, Fellow and Senior Fellow of the Zukunftskolleg, whose hospitality is gratefully acknowledged.

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Universität Konstanz, Zukunftskolleg/FB Mathematik und Statistik, D-78457 Konstanz, Germany.

Universiteit Antwerpen, Departement Wiskunde en Informatica, Middelheimlaan 1, B-2020 Antwerpen, Belgium

E-mail address: KarimJohannes.Becher@uantwerpen.be

Université Rouen Normandie, Laboratoire de Didactique André Revuz (LDAR, EA 4434), Université Artois, Université Cergy-Pontoise, Université Paris Diderot, Université Paris Est Créteil, F-76130 Mont-Saint-Aignan, France.

E-mail address: nicolas.grenier-boley@univ-rouen.fr

Université catholique de Louvain, ICTEAM Institute, Avenue G. Lemaître 4, Box L4.05.01, B-1348 Louvain-La-Neuve, Belgium.

E-mail address: jean-pierre.tignol@uclouvain.be