ON TOTALLY DECOMPOSABLE ALGEBRAS WITH INVOLUTION IN CHARACTERISTIC TWO

M. G. MAHMOUDI, A.-H. NOKHODKAR

Abstract. A necessary and sufficient condition for a central simple algebra with involution over a field of characteristic two to be decomposable as a tensor product of quaternion algebras with involution, in terms of its Frobenius subalgebras, is given. It is also proved that a bilinear Pfister form, recently introduced by A. Dolphin, can classify totally decomposable central simple algebras of orthogonal type.

Mathematics Subject Classification: 16W10, 16W25, 16K20, 11E39.

1. Introduction

An old result due to A. A. Albert states that every central simple algebra A of degree 4 which carries an involution of the first kind can be decomposed as a tensor product of two quaternion algebras (see [15, §16]). This result is no longer valid if A is of degree 8 by the examples given in [1] over fields of characteristic different from 2 and in [23] over fields of characteristic 2. In [1], it was also shown that if A is of degree 2^{n} over a field of characteristic different from 2, then A decomposes into a tensor product of quaternion algebras if and only if there exists a finite square-central subset of A (called a q-generating set) which satisfies some commuting properties. Over a field of particular cohomological dimension, it is known that central simple algebras which carry an involution of the first kind can be decomposed as a tensor product of quaternion algebras (see [12], and [4] for a characteristic 2 counterpart). In [3], a similar result was proved provided that the base field is of the u-invariant ⩽ 8.

A closely related problem is to determine the conditions under which a central simple algebra with involution (A, σ) is totally decomposable (i.e., (A, σ) decomposes as a tensor product of σ-invariant quaternion algebras). In [22], it was shown that if A is of degree 4 over a field of characteristic different from 2 and σ is of symplectic type, then A can be decomposed as a tensor product of two σ-invariant quaternion algebras. A proof of this result in characteristic 2 was given in [24], also a characteristic independent proof of this result and a criterion for decomposability in the case where σ is orthogonal can be found in [16]. A similar criterion for the unitary case of degree 4 and of arbitrary characteristic was derived in [13]. A cohomological invariant to detect decomposability for degree 8 algebras with symplectic involution over a field of characteristic different from 2 can be found in [9]. For the case of degree 8 algebras with orthogonal involution (A, σ) over a field of characteristic different from 2, a criterion for decomposability in terms of the Clifford algebra of (A, σ) can be found in [15, (42.11)], see also [25, (3.10)]. For the case where (A, σ) is split and of arbitrary degree 2^{n} over a field of characteristic different from 2, a decomposability criterion in term of higher degree invariants of a quadratic form q, to which σ is adjoint, can be found in [25].

Another relevant problem is to find invariants which classify central simple algebras with involution (A, σ) up to conjugation. Orthogonal involutions of degree ⩽ 4 can be classified by their Clifford algebras [15, §15], [17, §2]. A degree 4 central simple algebra with symplectic involution (A, σ) can be classified by a 3-fold Pfister form or an Albert form associated to σ, see [14] and [15, §16].

In this work we study the problems of decomposition and classification of central simple algebras with involution in the case of characteristic 2. In (4.5), we show that a central simple algebra with involution over a field of characteristic 2 is totally decomposable if and only if there exists a symmetric and self-centralizing subalgebra S = Φ(A, σ) of A such that (i) x^{2} ∈ F for every x ∈ S and (ii) dim_{F} S = 2^{r_{F} (S)}, where r_{F} (S) is the minimum rank of S. In the case where (A, σ) is totally decomposable central simple algebra with involution of orthogonal type we show that the aforementioned subalgebra Φ(A, σ), is unique up to isomorphism (see (5.10)). We prove the existence of a natural associative bilinear form 𝔰 on Φ(A, σ), isometric to a recently introduced bilinear Pfister form 𝔓𝔣(A, σ) in [7], thus providing a more intrinsic definition of 𝔓𝔣(A, σ) (see (5.5), (5.6)). In [7, (7.5)], it was shown that for every splitting field K of A, the involution σ_{K} on A_{K} is adjoint to the bilinear form 𝔓𝔣(A, σ)_{K} (see [7, (7.5)], compare [21, (5.1)]), and it was asked (see [7, (7.4)]) if 𝔓𝔣(A, σ) classify (A, σ) up to conjugation. Using the methods developed in the current work, we give in (6.5) an affirmative answer to this question.

2. Preliminaries

Let V be a finite dimensional vector space over a field F and let 𝔟 : V × V → F be a bilinear form. Let D_{F} (𝔟) = {𝔟(v, v) : v ∈ V and 𝔟(v, v) ≠ 0}. If K/F is a field extension, the extension of 𝔟 to V_{K} = V ⊗_{F} K is denoted by 𝔟_{K}.

The orthogonal sum and the tensor product of two bilinear forms 𝔟_{1} and 𝔟_{2} are denoted by 𝔟_{1} ⊥ 𝔟_{2} and 𝔟_{1} ⊗ 𝔟_{2} respectively. For α in F^{×}, the group of invertible elements of F, we use the notation ⟨α⟩ for the isometry class of the one-dimensional bilinear space (V, 𝔟) over F defined by 𝔟(u, v) = αuv. The bilinear form ⊥_{i=1}^{n}⟨α_{i}⟩ is denoted by ⟨α_{1}, ⋯, α_{n}⟩.

Let F be a field and let α_{1}, ⋯, α_{n} ∈ F^{×}. The 2^{n}-dimensional bilinear form ⟨1, α_{1}⟩ ⊗ ⋯ ⊗ ⟨1, α_{n}⟩ over F is called a bilinear n-fold Pfister form and is denoted by ⟨⟨α_{1}, ⋯, α_{n}⟩⟩. If 𝔟 is a bilinear Pfister form then there exists a bilinear form 𝔟′, uniquely determined up to isometry, such that 𝔟 = ⟨1⟩ ⊥ 𝔟′ (see [2, p. 16]). The form 𝔟′ is called the pure subform of 𝔟.

A quadratic form over F is a map q : V → F such that: (1) q(αv) = α^{2}q(v) for every α ∈ F and v ∈ V; (2) the map 𝔟_{q} : V × V → F defined by 𝔟_{q}(u, v) = q(u + v) − q(u) − q(v) for every u, v ∈ V is a bilinear form. We say that q is totally singular if 𝔟_{q}(u, v) = 0 for every u, v ∈ V. For α_{1}, ⋯, α_{n} ∈ F, the isometry class of the n-dimensional totally singular quadratic form (V, q) over F defined by q(v_{1}, …, v_{n}) = α_{1}v_{1}^{2} + ⋯ + α_{n}v_{n}^{2} is denoted by [α_{1}] ⊥ ⋯ ⊥ [α_{n}]. The Clifford algebra of a quadratic form (V, q) is denoted by C(V). We refer the reader to [8, Ch. II] for basic definitions and facts regarding Clifford algebras and quadratic and bilinear forms in arbitrary characteristic.

Let R be a ring. An additive map δ : R → R is called a derivation, if δ(ab) = aδ(b) + δ(a)b for every a, b ∈ R. For a ∈ R, the map δ_{a} : R → R defined by δ_{a}(x) = ax − xa is a derivation of R which is called the inner derivation induced by a.

All F -algebras considered in this work are supposed to be unital and associative. The reader is referred to [20, Ch. 12] for basic notions concerning central simple algebras. We just recall that the degree of a central simple algebra A is defined by deg_{F} A = \sqrt{dim_{F} A}. Also for a subalgebra B of A the centralizer of B in A and the center of B are denoted respectively by C_{A}(B) and Z(A).

TOTALLY DECOMPOSABLE ALGEBRAS WITH INVOLUTION IN CHARACTERISTIC 2 3

A finite dimensional algebra A over F is called a Frobenius algebra if A contains a hyperplane H that contains no nonzero left ideal of A; alternatively A is called a Frobenius F -algebra if there exists a nondegenerate bilinear form 𝔟 : A × A → F which is associative, in the sense that 𝔟(x, yz) = 𝔟(xy, z) for every x, y, z ∈ A. In this work, we use the following fundamental result about the properties of Frobenius subalgebras of central simple algebras:

Theorem 2.1. [11, (2.2.3)] Let A be a central simple algebra over a field F and let S be a commutative Frobenius subalgebra of A such that dim_{F} S = deg_{F} A.

(i) We have C_{A}(S) = S.

(ii) Every derivation of S into A can be extended to an inner derivation of A.

For further properties of Frobenius algebras see [11] .

The minimum rank of a finite dimensional F -algebra A which is denoted by r_{F} (A) is the minimum number r such that A can be generated as an F -algebra by r elements. Also the Loewy length of A which is denoted by ℓℓ(A) is defined as the smallest positive integer l such that J(A)^{l} = 0; in other words ℓℓ(A) is the nilpotency index of the Jacobson radical of A.

Let A be a central simple algebra over a field F. An involution on A is an antiautomorphism σ of A such that σ^{2} = id. The involution σ is called of the first kind if σ|_{F} = id. The set of alternating and symmetric elements of (A, σ) are defined as follows:

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

An involution σ of the first kind is said to be of symplectic type if over a splitting field of A, σ becomes adjoint to an alternating bilinear form. Otherwise, σ is said to be of orthogonal type. If char F = 2 and σ is of the first kind, then it can be shown that σ is of orthogonal type if and only if 1 ∉ Alt(A, σ), see [15, (2.6)]. The discriminant of an involution σ of orthogonal type is denoted by disc σ, see [15, (7.1)].

Let F be a field of characteristic 2. A quaternion algebra over F is a central simple F -algebra of degree 2. As an F -algebra, every quaternion algebra is generated by two elements u and v subject to the relations

u^{2} + u ∈ F, v^{2} ∈ F^{×} and uv + vu = v.

Furthermore {1, u, v, uv} is a basis of Q over F.

3. Totally singular conic Frobenius algebras

Definition 3.1. In analogy with [10], we call an algebra R over a field F a totally singular conic algebra if x^{2} ∈ F for every x ∈ R.

Remark 3.2. Let F be a field of characteristic 2 and let R be a finite dimensional totally singular conic F -algebra. It follows immediately that

(i) R is a local commutative algebra and its unique maximal ideal is 𝔪 = {x ∈ R : x^{2} = 0}.

(ii) For every u ∈ R \ F, the subalgebra F [u] is a field if and only if u^{2} ∉ F^{2} = {x^{2} : x ∈ F }.

Remark 3.3. A local commutative algebra is a Frobenius algebra if and only if it has a unique minimal ideal, see [11, (2.1.3)]. In particular for a finite dimensional totally singular conic algebra, being a Frobenius algebra is a purely ring theoretic property and dose not depend on the base field.

Lemma 3.4. Let R be a finite dimensional totally singular conic algebra over a field F. If dim_{F} R = 2^{r_{F} (R)}, then R is a Frobenius algebra.

Proof. Set n = r_{F} (R) and write R = F [u_{1}, ⋯, u_{n}] for some u_{1}, ⋯, u_{n} ∈ R. The F -algebra homomorphisms f_{i} : F [u_{i}] → R defined by f_{i}(u_{i}) = u_{i}, i = 1, ⋯, n, induce a surjective F -algebra homomorphism f : F [u_{1}] ⊗ ⋯ ⊗ F [u_{n}] → R. By dimension count f is an isomorphism. We know that single generated algebras (i.e., algebras of the form F [u]) and the tensor product of Frobenius algebras are Frobenius (see [11, (2.1.4)] and [11, (2.1.2)]), hence R is a Frobenius F -algebra. □

Remark 3.5. The converse of (3.4) is not necessarily true. Here we construct a counter example. Let F be a field of characteristic 2 and for n ≥ 4, let R_{n} be the n-dimensional algebra over F with the basis {1, u_{1}, ⋯, u_{n−1}} subject to the relations

u_{i}^{2} = u_{1}u_{i} = 0, 1 ≤ i ≤ n − 1,
(1)  u_{i}u_{j} = u_{1}, 2 ≤ i ≠ j ≤ n − 1.

It is easy to see that the above relations imply that u_{1}u_{i} = u_{i}u_{1} for every i. It follows that R_{n} is a totally singular conic algebra with the unique maximal ideal 𝔪 = F u_{1} + ⋯ + F u_{n−1}. In particular for every element x ∈ R_{n} one can write x = a + m, where a ∈ F and m ∈ 𝔪. Set I := R_{n}u_{1} = F u_{1}. Then I is a minimal ideal of R_{n}. Let x, y ∈ R_{n} and write x = a + m and y = b + m′ where a, b ∈ F and m, m′ ∈ 𝔪. By (1) we have mm′ ∈ I. So there exist c ∈ F such that

xy = ay + bx + cu_{1} − ab. (2)

Let r = r_{F} (R_{n}) and write R_{n} = F [v_{1}, ⋯, v_{r}] for some v_{1}, ⋯, v_{r} ∈ R_{n}. Set S = F v_{1} + ⋯ + F v_{r} ⊆ R_{n}. By (2) every monomial in terms of v_{1}, ⋯, v_{r} belongs to the subspace S + F u_{1} + F. Since these monomials generate R_{n} as an F -algebra and dim_{F} R_{n} = n, we obtain r ≥ n − 2. On the other hand R_{n} is generated as an F -algebra by the elements u_{2}, ⋯, u_{n−1}, so r_{F} (R_{n}) = r = n − 2.

Now suppose that n is even. We claim that I is the unique minimal ideal of R_{n}. Let J ≠ {0} be an ideal of R_{n} and let 0 ≠ x ∈ J. As J ⊆ 𝔪 we have

x = ∑_{i=1}^{n−1} a_{i}u_{i}, (3)

for a_{1}, ⋯, a_{n−1} ∈ F. Set b_{i} = (∑_{j=2}^{n−1} a_{j}) − a_{i}, i = 2, ⋯, n − 1. Multiplying (3) by u_{i} we get b_{i}u_{1} ∈ J, i = 2, ⋯, n − 1. If b_{i} ≠ 0 for some 2 ≤ i ≤ n − 1 then u_{1} ∈ J and I ⊆ J. Otherwise b_{2} = ⋯ = b_{n−1} = 0 which leads to a system of linear equations with respect to a_{2}, ⋯, a_{n−1}. As n is even (and char F = 2) it is easy to see that the only solution of this system is the trivial solution, i.e., a_{2} = ⋯ = a_{n−1} = 0. Since x ≠ 0 we obtain a_{1} ≠ 0, so again u_{1} ∈ J, i.e., I ⊆ J. So the claim is proved and by (3.3), R_{n} is a Frobenius algebra. For every even integer n ≥ 6 the algebra R_{n} is a totally singular conic algebra which is Frobenius, but dim_{F} R_{n} = n ≠ 2^{n−2} = 2^{r_{F} (R_{n})}. Also even if dim_{F} R is a power of 2, the converse of (3.4) is not true; take n = 2^{k}, k ≥ 3 and R = R_{n}.

Remark 3.6. Let R be a finite dimensional totally singular conic algebra over a field F of characteristic 2 and let L ⊇ F be a subfield of R. If R is a field, then r_{L}(R) + r_{F} (L) = r_{F} (R). This fact is an easy consequence of the multiplication formula [R : F] = [R : L][L : F] and the fact that [K : F] = 2^{r_{F} (K)} for every subfield K ⊇ F of R.

Lemma 3.7. Let R be a finite dimensional totally singular conic algebra over a field F of characteristic 2 and let 𝔪 be its unique maximal ideal mentioned in (3.2). Set r = r_{F} (R/𝔪) and n = r_{F} (R).

(i) If K ⊇ F is a maximal subfield of R, then the residue field R/𝔪 and K are isomorphic as F -algebras. In particular K is unique up to F -algebra isomorphism and [K : F] = dim_{F} R/𝔪 = 2^{r}. Also for every x ∈ R we have x^{2} ∈ K^{2}.

(ii) There exist a maximal subfield K ⊇ F of R and u_{1}, ⋯, u_{n−r} ∈ 𝔪 such

that R = K[u_{1}, ⋯, u_{n−r}]. (iii) We have ℓℓ(R) ≤ r_{F} (R) − r_{F} (R/𝔪) + 1.

Proof. (i) Since F ⊆ K ⊆ R, R is a finite dimensional totally singular conic Kalgebra as well. Since K is maximal, using (3.2 (ii)) we have x^{2} ∈ K^{2} for every x ∈ R. Consider the map φ : K → R/𝔪 defined by φ(x) = x + 𝔪. Clearly φ is an injective F -algebra homomorphism. We show that φ is surjective. Let x+𝔪 ∈ R/𝔪, where x ∈ R. As x^{2} ∈ K^{2}, there exists y ∈ K such that x^{2} = y^{2} ∈ K^{2}, i.e., (y + x)^{2} = 0. So we have y + x ∈ 𝔪 which implies that φ(y) = y + 𝔪 = x + 𝔪.

(ii) By induction on n, we first prove that there exist a maximal subfield K ⊇ F of R and u_{1}, ⋯, u_{n−r} ∈ R such that R = K[u_{1}, ⋯, u_{n−r}]. Choose v_{1}, ⋯, v_{n} ∈ R such that R = F [v_{1}, ⋯, v_{n}]. If v_{i}^{2} ∈ F^{2}, for every i = 1, ⋯, n, then for every u ∈ R we obtain u^{2} ∈ F^{2}, so (3.2 (ii)) implies that every maximal subfield K ⊇ F of R reduces to F, i.e., r = 0 and we are done. Otherwise (by re-indexing if necessary) we may assume that v_{1}^{2} ∉ F^{2}. Then L := F [v_{1}] is a quadratic extension of F and R = L[v_{2}, ⋯, v_{n}]. By (3.3), R is a Frobenius L-algebra. We have r_{L}(R) = n − 1, also (3.6) implies that r_{L}(R/𝔪) = r_{F} (R/𝔪) − r_{F} (L) = r − 1. So by induction hypothesis there exist a maximal subfield K ⊇ L of R and u_{1}, ⋯, u_{n−r} ∈ R such that R = K[u_{1}, ⋯, u_{n−r}].

Since K is maximal we have u_{i}^{2} ∈ K^{2}, i = 1, ⋯, n−r. Replacing u_{i} with u_{i} +α_{i} for some α_{i} ∈ K, we may assume that u_{i}^{2} = 0, i = 1, ⋯, n − r.

(iii) By the previous part, there exist a maximal subfield K ⊇ F and u_{1}, ⋯, u_{n−r} ∈ 𝔪 such that R = K[u_{1}, ⋯, u_{n−r}]. Consider arbitrary elements x_{1}, ⋯, x_{n−r+1} ∈ 𝔪. Since K ∩ 𝔪 = {0}, every x_{i} can be written as

x_{i} = ∑_{\begin{matrix}1≤l≤n−r \\ 1≤i_{1}<···<i_{l}≤n−r \end{matrix}} α_{i_{1}···i_{l}}u_{i_{1}} ⋯ u_{i_{l}}, α_{i_{1}···i_{l}} ∈ K.

So every monomial in the expansion of x_{1} ⋯ x_{n−r+1} in terms of u_{1}, ⋯, u_{n−r} has two identical u_{i}’s. As u_{i}^{2} = 0 we have x_{1} ⋯ x_{n−r+1} = 0. Thus we obtain 𝔪^{n−r+1} = 0, i.e., ℓℓ(R) ≤ r_{F} (R) − r_{F} (R/𝔪) + 1. □

The following result shows that (3.6) is also true for every finite dimensional totally singular conic algebra:

Corollary 3.8. Let R be a finite dimensional totally singular conic algebra over a field F of characteristic 2. If L ⊇ F is a subfield of R then r_{L}(R)+r_{F} (L) = r_{F} (R).

Proof. If L = F (in other words r_{F} (L) = 0) the result trivially holds. So suppose that r_{F} (L) ≥ 1. We obviously have r_{F} (R) ≤ r_{L}(R) + r_{F} (L). So it is enough to show that r_{F} (R) ≥ r_{L}(R) + r_{F} (L). We prove this for the case where r_{F} (L) = 1. The general case follows from induction. Let 𝔪 be the unique maximal ideal of R mentioned in (3.2), r = r_{F} (R/𝔪) and n = r_{F} (R). By (3.7 (ii)) there exist a maximal subfield K ⊇ F of R and u_{1}, ⋯, u_{n−r} ∈ 𝔪 such that R = K[u_{1}, ⋯ , u_{n−r}]. Write L = F [u] for some u ∈ R with u^{2} ∈ F^{×}\F^{×2}. Extend L to a maximal subfield K′ of R. By (3.7 (i)) we have K′ ≃ K, so there exists v_{1} ∈ K such that v_{1}^{2} = u^{2} ∈ F^{×}\F^{×2}. Set m := v_{1}+u ∈ R. Since m^{2} = (v_{1}+u)^{2} = 0, we have m ∈ 𝔪. As r_{F} (K) = r and K is a field, by (3.6) we have r_{F [v_{1}]}(K) = r − 1. So there exist v_{2}, ⋯, v_{r} ∈ K such that K = F [v_{1}, ⋯, v_{r}]. Then R = F [v_{1}, ⋯, v_{r}, u_{1}, ⋯, u_{n−r}]. Set S = L[v_{2}, ⋯, v_{r}, u_{1}, ⋯, u_{n−r}]. We claim that S = R which implies that r_{L}(R) ≤ n − 1 = r_{F} (R) − 1. It is enough to show that v_{1} ∈ S. As R = K[u_{1}, ⋯ , u_{n−r}], one can write

(4)  m = ∑_{\begin{matrix}1≤l≤n−r \\ 1≤i_{1}<···<i_{l}≤n−r \end{matrix}} α_{i_{1}···i_{l}}u_{i_{1}} ⋯ u_{i_{l}}, α_{i_{1}···i_{l}} ∈ K.

Every α_{i_{1}···i_{l}} ∈ K = F [v_{1}, ⋯, v_{r}] can be written as

α_{i_{1}···i_{l}} = β_{i_{1}···i_{l}} + γ_{i_{1}···i_{l}}v_{1},

where β_{i_{1}···i_{l}}, γ_{i_{1}···i_{l}} ∈ F [v_{2}, ⋯, v_{r}] ⊆ S. So by (4) there exist s, s′ ∈ S such that m = v_{1}s + s′. As u_{1}, ⋯, u_{n−r} ∈ 𝔪 and 𝔪 is an ideal, we obtain s, s′ ∈ 𝔪, so s, s′ ∈ S ∩ 𝔪. We obtain therefore u = v_{1} + m = v_{1}(1 + s) + s′, so (1 + s)u = v_{1} + (1 + s)s′, i.e., v_{1} = (1 + s)(u + s′) ∈ S. □

Remark 3.9. The statement of (3.7 (ii)) can be strengthened as follows: “For every maximal subfield K ⊇ F of R there exist u_{1}, ⋯, u_{n−r} ∈ 𝔪 such that R = K[u_{1}, ⋯, u_{n−r}].” In fact by (3.8) we have r_{K}(R) = n − r. So there exist u_{1}, ⋯, u_{n−r} ∈ R such that R = K[u_{1}, ⋯, u_{n−r}]. Since K is maximal we have u_{i}^{2} ∈ K^{2} for i = 1, ⋯, n − r. Replacing u_{i} with u_{i} + α_{i} for some α_{i} ∈ K, we may assume that u_{1}^{2} = ⋯ = u_{n−r}^{2} = 0.

Definition 3.10. Let R be a finite dimensional totally singular conic algebra over a field F of characteristic 2 and let 𝔪 be its unique maximal ideal. We say that R is a ρ-generated algebra if r_{F} (R) = ρ(R) where ρ(R) = ℓℓ(R) + r_{F} (R/𝔪) − 1.

Proposition 3.11. Let F be a field of characteristic 2 and let R be a finite dimensional totally singular conic F -algebra. Then R is ρ-generated if and only if dim_{F} R = 2^{r_{F} (R)}. In particular every ρ-generated totally singular conic algebra is a Frobenius algebra.

Proof. By (3.2), R is a local commutative algebra. Let 𝔪 be the unique maximal ideal of R, r = r_{F} (R/𝔪) and n = r_{F} (R).

Suppose that R is a ρ-generated algebra. As r_{F} (R) = n we have dim_{F} R ≤ 2^{n}. So it is enough to show that dim_{F} R ≥ 2^{n}. Let K ⊇ F be a maximal subfield of R and write K = F [u_{1}, ⋯, u_{r}] for some u_{1}, ⋯, u_{r} ∈ K. Since ℓℓ(R) = n − r + 1 we have 𝔪^{n−r} ≠ 0, so there exist v_{1}, ⋯, v_{n−r} ∈ 𝔪 such that v_{1} ⋯ v_{n−r} ≠ 0. We show that

dim_{K} K[v_{1}, ⋯, v_{n−r}] = 2^{n−r}, (5)

which concludes that

dim_{F} R ≥ dim_{F} K · dim_{K} K[v_{1}, ⋯, v_{n−r}] = 2^{n}.

In order to prove (5), we claim that the set

W = {1} ∪ {v_{i_{1}} ⋯ v_{i_{l}} : 1 ≤ i_{1} < ⋯ < i_{l} ≤ n − r, 1 ≤ l ≤ n − r},

is linearly independent over K. Suppose that

∑_{\begin{matrix}1≤l≤n−r \\ 1≤i_{1}<···<i_{l}≤n−r \end{matrix}} α_{i_{1}···i_{l}}v_{i_{1}} ⋯ v_{i_{l}} = α, α, α_{i_{1}···i_{l}} ∈ K, (6)

where at least one of the above terms is nonzero and the number of nonzero terms is minimal. Since the left side of (6) belongs to 𝔪 we have α = 0. As v_{j}^{2} = 0, j = 1, ⋯, n − r, multiplying the equality (6) by v_{j} implies that either v_{j} does not appear in the above sum or appears in all terms. It follows that the only nonzero term of the left side of (6) is a multiple of v_{j_{1}} ⋯ v_{j_{l}} for some 1 ≤ j_{1} < ⋯ < j_{l} ≤ n−r and 1 ≤ l ≤ n − r which contradicts the assumption v_{1} ⋯ v_{n−r} ≠ 0. So the claim is proved and dim_{K} K[v_{1}, ⋯, v_{n−r}] = 2^{n−r}.

Conversely suppose that dim_{F} R = 2^{n}. As dim_{F} K = 2^{r}, we have dim_{K} R = 2^{n−r}. By (3.9) there exist u_{1}, ⋯, u_{n−r} ∈ 𝔪 such that R = K[u_{1}, ⋯, u_{n−r}]. It follows that u_{1} ⋯ u_{n−r} ≠ 0, i.e., 𝔪^{n−r} ≠ 0. So ℓℓ(R) ≥ n − r + 1 and thanks to (3.7 (iii)), R is a ρ-generated algebra. The last statement of the result follows from (3.4). □

Remark 3.12. For n = 2^{k}, k ≥ 3, the totally singular conic algebra R_{n} constructed in (3.5) is a Frobenius algebra which is not ρ-generated, because dim_{F} R_{n} ≠ 2^{r_{F} (R_{n})}. So the converse of the second statement of (3.11) does not hold.

The next result follows from (3.11) and the standard properties of tensor product.

Corollary 3.13. Let R and R′ be two finite dimensional ρ-generated totally singular conic algebras over a field F of characteristic 2. Then R ⊗_{F} R′ is also a ρ-generated totally singular conic F -algebra.

Lemma 3.14. Let (V, q) be a quadratic form over a field F of characteristic 2 and let f : V → C(V) be an F -linear map such that f(v) ∈ Z(C(V)) for every v ∈ V. Then the map f can be uniquely extended to an F -derivation δ : C(V) → C(V).

Proof. Let T (V) be the tensor algebra of V with the canonical map ¯ : T (V) → C(V). Let g : T (V) → C(V) be the linear map induced by g(1) = 0 and

g(u_{1} ⊗ ⋯ ⊗ u_{n}) = f(u_{1})\overline{u_{2}} ⋯ \overline{u_{n}} + \overline{u_{1}}f(u_{2})\overline{u_{3}} ⋯ \overline{u_{n}}

+ ⋯ + \overline{u_{1}} ⋯ \overline{u_{n−1}}f(u_{n}), (7)

for u_{1}, ⋯, u_{n} ∈ V. For every w_{1}, w_{2} ∈ T (V) we have

g(w_{1} ⊗ w_{2}) = \overline{w_{1}}g(w_{2}) + g(w_{1})\overline{w_{2}}. (8)

Let I be the ideal of T (V) generated by the elements of the form v ⊗ v − q(v) for v ∈ V. We claim that g(I) = 0. For every w ∈ T (V) and v ∈ V we have

g(w ⊗ (v ⊗ v − q(v))) = \overline{w}g(v ⊗ v − q(v)) + g(w)(\overline{v ⊗ v − q(v)})
= \overline{w}g(v ⊗ v − q(v)) + 0 = \overline{w}(g(v ⊗ v) − g(q(v)))
= \overline{w}g(v ⊗ v) = \overline{w}(\overline{v}f(v) + f(v)\overline{v}) = 0,

where in the last equality we used the assumption f(v) ∈ Z(C(V)). Similarly for every w ∈ T (V) and v ∈ V we have g((v ⊗ v − q(v)) ⊗ w) = 0. So g(I) = 0 and g can be factored through C(V), i.e., g induces a map δ : C(V) → C(V). By (8), for every w_{1}, w_{2} ∈ C(V) we have δ(w_{1}w_{2}) = w_{1}δ(w_{2}) + δ(w_{1})w_{2}, so δ is a derivation.

The uniqueness of δ follows from the fact that V generates C(V) as an F - algebra. □

Lemma 3.15. Let R be a finite dimensional algebra over a field F of characteristic 2 and let n = r_{F} (R). Then R is a ρ-generated totally singular conic F -algebra if and only if there exists a totally singular quadratic form (V, q) of dimension n over F such that R ≃ C(V). In addition V can be chosen as the vector space generated by every generating subset {u_{1}, ⋯, u_{n}} of R (as F -algebra) with q(v) = v^{2} ∈ F for every v ∈ V.

Proof. First suppose that R is a ρ-generated totally singular conic F -algebra. Write R = F [u_{1}, ⋯, u_{n}] for some u_{1}, ⋯, u_{n} ∈ R. Set α_{i} = u_{i}^{2} ∈ F, i = 1, ⋯, n and V = F u_{1} + ⋯ + F u_{n} ⊆ R. Define the map q : V → F via q(v) = v^{2} ∈ F. For every u, v ∈ V we have q(u + v) − q(u) − q(v) = (u + v)^{2} − u^{2} − v^{2} = 0. So q is a totally singular quadratic form. Consider the inclusion map i : V ↪ R. The map i is compatible with q and can be extended to an F -algebra homomorphism φ : C(V) → R. As R = F [u_{1}, ⋯, u_{n}], φ is surjective. Also using (3.11), we have dim_{F} R = 2^{n} = dim_{F} C(V), so φ is an isomorphism.

Conversely suppose that R ≃ C(V) for a totally singular quadratic form (V, q) over F. Let {v_{1}, ⋯, v_{n}} be a basis of V over F. As (V, q) is totally singular, C(V) is a totally singular conic F -algebra. Also as an F -algebra, C(V) is generated by the set {v_{1}, ⋯, v_{n}}, i.e., r_{F} (C(V)) ≤ n. Since dim_{F} C(V) = 2^{n} we obtain n = r_{F} (C(V)). So dim_{F} C(V) = 2^{r_{F} (C(V))} and by (3.11), C(V) (and therefore R) is ρ-generated. □

Corollary 3.16. Let R and R′ be two finite dimensional ρ-generated totally singular conic algebras over a field F of characteristic 2 with respective maximal subfields K and K′. Then there exists an F -algebra isomorphism R ≃ R′ if and only if dim_{F} R = dim_{F} R′ and K ≃ K′ as F -algebras.

Proof. First suppose that dim_{F} R = dim_{F} R′ and K ≃ K′. By (3.11) we have dim_{F} R = 2^{r_{F} (R)} and dim_{F} R′ = 2^{r_{F} (R′)}. So dim_{F} R = dim_{F} R′ = 2^{n}, where n = r_{F} (R) = r_{F} (R′). Let 𝔪 and 𝔪′ be respectively the unique maximal ideals of R and R′ mentioned in (3.2) and set r = r_{F} (R/𝔪). As K ≃ K′, by (3.7 (i)) we have R/𝔪 ≃ R′/𝔪′ as F -algebras. So r_{F} (R′/𝔪′) = r. Also by (3.9) there exist u_{1}, ⋯, u_{n−r} ∈ 𝔪 and u′_{1}, ⋯, u′_{n−r} ∈ 𝔪′ such that R = K[u_{1}, ⋯, u_{n−r}] and R′ = K′[u′_{1}, ⋯, u′_{n−r}]. As dim_{F} R = 2^{n} and dim_{F} K = 2^{r}, we have dim_{K} R = 2^{n−r}. So the set {u_{1}, ⋯, u_{n−r}} is linearly independent over K. Set V = Ku_{1} + ⋯ + Ku_{n−r} and V′ = K′u′_{1} + ⋯ + K′u′_{n−r}. Define the quadratic forms q on V and q′ on V′ via q(v) = v^{2} and q′(v′) = v′^{2}. Since u_{i} ∈ 𝔪 and u′_{i} ∈ 𝔪′, i = 1, ⋯, n − r, q and q′ are zero maps. So C(V) ≃ C(V′) as K-algebras. On the other hand by (3.15) we have R ≃ C(V) and R′ ≃ C(V′) as F -algebras, so R ≃ R′ and we are done. The converse is trivial. □

Proposition 3.17. Let R be a finite dimensional totally singular conic algebra over a field F of characteristic 2 and let n = r_{F} (R). Consider a subset B := {u_{1}, ⋯, u_{n}} ⊆ R which generates R as an F -algebra. Then the following statements are equivalent:

(i) The algebra R is ρ-generated.

(ii) Every map f : B → R can be uniquely extended to an F -derivation δ : R →

R. (iii) Every map f : B → A satisfying f(u_{i})^{2} = u_{i}^{2} ∈ F, i = 1, ⋯, n, can be

uniquely extended to an F -algebra homomorphism φ : R → A.

Proof. Let V = F u_{1} + ⋯ + F u_{n} and let q : V → F be the quadratic form defined by q(v) = v^{2}. By (3.15), R is ρ-generated if and only if R ≃ C(V). So the equivalence (i) ⇔ (iii) follows from the universal property of Clifford algebras and the implication (i) ⇒ (ii) follows from (3.14). (ii) ⇒ (i): Let K be a maximal subfield of R and set r = r_{F} (K). Let 𝔪 be the unique maximal ideal of R mentioned in (3.2). By (3.9) there exist v_{1}, ⋯, v_{n−r} ∈ 𝔪 such that R = K[v_{1}, ⋯, v_{n−r}]. We claim that v_{1} ⋯ v_{n−r} ≠ 0. If v_{1} ⋯ v_{n−r} = 0, then there exists a minimal number l ≤ n − r and 1 ≤ i_{1} < i_{2} < ⋯ < i_{l} ≤ n − r such that

v_{i_{1}} ⋯ v_{i_{l}} = 0. (9)

Choose v_{n−r+1}, ⋯, v_{n} ∈ R such that K = F [v_{n−r+1}, ⋯, v_{n}], so {v_{1}, ⋯, v_{n}} generates R as an F -algebra. Let f : {v_{1}, ⋯, v_{n}} → R be the map defined by f(v_{i_{1}}) = 1 and f(v_{j}) = 0 for every j ≠ i_{1}. By the hypothesis, f can be extended to a derivation δ on R. Note that l ≥ 2, because all v_{i}’s are nonzero. Applying δ to (9) we obtain v_{i_{2}} ⋯ v_{i_{l}} = 0, which contradicts the minimality of l. So the claim is proved and we have 𝔪^{n−r} ≠ 0. It follows that ℓℓ(R) ≥ n − r + 1 and thanks to (3.7 (iii)), R is a ρ-generated algebra. □

4. Decomposability in terms of Frobenius subalgebras

Remarks 4.1. Let (Q, σ) be a quaternion algebra with involution over a field F of characteristic 2.

(i) There exists u ∈ Sym(Q, σ) \ F such that u^{2} ∈ F^{×}. In fact if σ is of symplectic type and u ∈ Sym(Q, σ), then u has trivial reduced trace (see [15, pp. 25-26]), so u^{2} ∈ F. Replacing u with u + 1 we may assume that u^{2} ∈ F^{×}. If σ is of orthogonal type, Alt(Q, σ) is one dimensional and by [15, (2.8 (2))] every nonzero element in Alt(Q, σ) is invertible. So for every 0 ≠ u ∈ Alt(Q, σ) we have u^{2} = Nrd_{Q}(u) ∈ F^{×}, where Nrd_{Q}(u) is the reduced norm of u in Q. (ii) If t is the transpose involution, (Q, σ) ≃ (M_{2}(F), t) if and only if σ is of orthogonal type and disc σ is trivial, see [15, (7.4)].

The proof of the following result uses the method of that of [16, (3.7)] and [19]^{1}. The technique was only applied for biquaternion algebras, but the main idea of the general case was present there.

Lemma 4.2. Let (A, σ) be a central simple algebra with involution over a field F of characteristic 2. Suppose that there exists a totally singular conic subalgebra S ⊆ Sym(A, σ) such that C_{A}(S) = S and dim_{F} S = 2^{r}, where r = r_{F} (S).

(i) There exists a quaternion F -subalgebra Q ⊆ A such that σ(Q) = Q.

(ii) If S = F [u_{1}, ⋯, u_{r}], where u_{1}, ⋯, u_{r} ∈ S, then the quaternion subalgebra

Q in part (i) can be chosen so that F [u_{2}, ⋯, u_{r}] ⊆ C_{A}(Q). (iii) If σ is of orthogonal type, then the quaternion subalgebra Q in part (i) can

be chosen so that S ∩ Alt(Q, σ|_{Q}) has an invertible element.

Proof. (i) By (3.4), S is a Frobenius algebra. Write S = F [u_{1}, ⋯, u_{r}], where u_{i} ∈ S. Replacing u_{i} with u_{i} + 1 if necessary, which doesn’t change F [u_{2}, ⋯, u_{r}] and S, we may assume that

0 ≠ u_{i}^{2} ∈ F, i = 1, ⋯, r. (10)

By (3.17), there exists an F -derivation δ of S induced by δ(u_{1}) = u_{1} and δ(u_{i}) = 0, i = 2, ⋯, r. By (2.1 (ii)), δ extends to an inner derivation δ_{ξ} of A for some ξ ∈ A. As δ^{2} = δ on S, the element ξ^{2} + ξ commutes with S, so ξ^{2} + ξ ∈ C_{A}(S) = S. Let η = ξ^{2} ∈ A. We have

η^{2} + η = ξ^{4} + ξ^{2} = (ξ^{2} + ξ)^{2} ∈ F. (11)

Since ξ and ξ^{2} induce the same derivation δ on S, we have

ηu_{1} + u_{1}η = ξu_{1} + u_{1}ξ = δ(u_{1}) = u_{1}. (12)

For x ∈ S the relations

δ_{σ(ξ)}(x) = σ(ξ)x − xσ(ξ) = σ(ξ)σ(x) − σ(x)σ(ξ) = σ(xξ − ξx) = xξ − ξx,

imply that the elements ξ and σ(ξ) also induce the same derivation δ on S. Thus σ(ξ) = ξ + s for some s ∈ S and

σ(η) = σ(ξ)^{2} = (ξ + s)^{2} = η + δ(s) + s^{2}, (13)

with δ(s) ∈ S, since S is stable under δ. Also

s^{2} ∈ F, (14)

since S is totally singular conic algebra. Now consider two cases. Case 1. If δ(s) ∈ F, then (13) implies that η + σ(η) ∈ F. By (10), (11) and (12), u_{1} and η generate a quaternion algebra Q, which is invariant under σ, because η + σ(η) ∈ F. Case 2. If δ(s) ∈ S \ F, we obtain

1We are grateful to Professor M.-A. Knus for making [19] available to us.

δ(s) = δ^{2}(s) = δ_{ξ^{2}}(δ(s)) = ηδ(s) + δ(s)η. (15)

Then by (11), (14) and (15), η and δ(s) generate a quaternion subalgebra Q of A, which is invariant under σ thanks to (13). (ii) Since δ(u_{2}) = ⋯ = δ(u_{r}) = 0, in both cases above F [u_{2}, ⋯, u_{r}] commutes with η, thus F [u_{2}, ⋯, u_{r}] ⊆ C_{A}(Q). (iii) Suppose that σ is of orthogonal type. In case 1 above, as η + σ(η) ∈ F, we obtain η = σ(η). Therefore using (12) we get ηu_{1} + σ(ηu_{1}) = ηu_{1} + u_{1}η = u_{1}, so u_{1} ∈ S ∩ Alt(Q, σ|_{Q}). Similarly in the case 2, we have δ(s) + s^{2} ∈ S ∩ Alt(Q, σ|_{Q}) by (13). So in both cases there exists a nonzero element in S ∩ Alt(Q, σ|_{Q}) which is invertible by [15, (2.8 (2))]. □

Corollary 4.3. Let (A, σ) be a central simple algebra of degree 2^{n} with involution of orthogonal type over a field F of characteristic 2. Suppose that there exists a totally singular conic subalgebra S ⊆ Sym(A, σ) such that C_{A}(S) = S and dim_{F} S = 2^{r_{F} (S)}. Then for i = 1, ⋯, n, there exists a σ-invariant quaternion algebra Q_{i} ⊆ A and an invertible element v_{i} ∈ S ∩ Alt(Q_{i}, σ|_{Q_{i}}) such that (A, σ) ≃ ⨂_{i=1}^{n}(Q_{i}, σ|_{Q_{i}}) and S = F [v_{1}, ⋯, v_{n}]. In particular dim_{F} S = 2^{n}.

Proof. Set r = r_{F} (S). Write S = F [u_{1}, ⋯, u_{r}] for some u_{1}, ⋯, u_{r} ∈ S and set S′ = F [u_{2}, ⋯, u_{r}]. By (4.2 (i)) there exists a quaternion subalgebra Q_{1} of A such that (A, σ) ≃ (Q_{1}, σ|_{Q_{1}}) ⊗ (B, τ), where B = C_{A}(Q_{1}) and τ = σ|_{B}. Also by (4.2 (ii)), Q_{1} can be chosen such that S′ ⊆ C_{A}(Q_{1}) = B. We have

F [u_{1}] ⊗ S′ ≃ S = C_{A}(S) ≃ C_{Q_{1}}(F [u_{1}]) ⊗ C_{B}(S′) ≃ F [u_{1}] ⊗ C_{B}(S′),

so dim_{F} C_{B}(S′) = dim_{F} S′. As S′ is commutative we obtain C_{B}(S′) = S′. Since dim_{F} S′ = 2^{r_{F} (S′)}, the induction can be applied to B and we obtain (A, σ) ≃ ⨂_{i=1}^{n}(Q_{i}, σ_{i}) where (Q_{i}, σ_{i}) is a quaternion algebra over F with involution.

By (4.2 (iii)) for every i there exists an invertible element v_{i} ∈ S ∩Alt(Q_{i}, σ|_{Q_{i}}). We claim that S = F [v_{1}, ⋯, v_{n}]. Set S′′ = F [v_{1}, ⋯, v_{n}] ⊆ S. Then dim_{F} S′′ = 2^{n} = 2^{r_{F} (S′′)}, so by (3.4) and (2.1 (i)), S′′ is a Frobenius algebra and C_{A}(S′′) = S′′. As S′′ ⊆ S and C_{A}(S) = S, we have S′′ = S. □

Remark 4.4. The converse of (4.3) is trivially true. In fact let (A, σ) = ⨂_{i=1}^{n}(Q_{i}, σ_{i}) be a totally decomposable algebra with involution over a field F of characteristic 2. By (4.1) there exists v_{i} ∈ Sym(Q_{i}, σ_{i}) such that v_{i}^{2} ∈ F^{×}. Set S = F [v_{1}, ⋯, v_{n}] ⊆ Sym(A, σ). Then S is a totally singular conic F -algebra and dim_{F} S = 2^{n} = 2^{r_{F} (S)}. By (3.4) and (2.1 (i)), we have C_{A}(S) = S.

Theorem 4.5. Let (A, σ) be a central simple algebra of degree 2^{n} with involution over a field F of characteristic 2. Then the following statements are equivalent:

(i) (A, σ) is totally decomposable.

(ii) There exists a ρ-generated totally singular conic F -algebra S ⊆ Sym(A, σ)

such that C_{A}(S) = S. Furthermore if σ is of orthogonal type then for every F -algebra S satisfying (ii), we have necessarily S ⊆ Alt(A, σ) + F. More precisely there exist v_{1}, ⋯, v_{n} ∈ Alt(A, σ) ∩ A^{∗} such that S = F [v_{1}, ⋯, v_{n}] and v_{i_{1}} ⋯ v_{i_{l}} ∈ Alt(A, σ) for every 1 ≤ i_{1} < ⋯ < i_{l} ≤ n and 1 ≤ l ≤ n. Thus, if σ is of orthogonal type the statement (ii) can be replaced by the following:

(iii) There exists a ρ-generated totally singular conic F -algebra S ⊆ Alt(A, σ) + F such that C_{A}(S) = S.

Finally if S is any subalgebra of A satisfying the condition (ii) or (iii) then r_{F} (S) = n and dim_{F} S = deg_{F} A = 2^{n}.

Proof. The equivalence (i) ⇔ (ii) follows from (3.11), (4.3) and (4.4).

If σ is of orthogonal type, then by (4.3) for every i there exists an invertible element v_{i} ∈ S ∩ Alt(Q_{i}, σ_{i}) such that S = F [v_{1}, ⋯, v_{n}]. Choose an element w_{i} ∈ Q_{i} such that v_{i} = w_{i} − σ(w_{i}), i = 1, ⋯, n. Then v_{i_{1}} ⋯ v_{i_{l}} = w_{i_{1}}v_{i_{2}} ⋯ v_{i_{l}} − σ(w_{i_{1}}v_{i_{2}} ⋯ v_{i_{l}}) ∈ Alt(A, σ) for every 1 ≤ i_{1} < ⋯ < i_{l} ≤ n and 1 ≤ l ≤ n.

The last statement of the result follows from (4.3). □

5. A new description of the Pfister invariant

Definition 5.1. Let (A, σ) be a totally decomposable algebra with involution of orthogonal type over a field F of characteristic 2. By (4.5 (ii)) there exists a ρ-generated totally singular conic algebra S ⊆ Sym(A, σ) such that C_{A}(S) = S. By (4.5 (iii)) we have necessarily S ⊆ Alt(A, σ) + F and there exist v_{1}, ⋯, v_{n} ∈ Alt(A, σ) ∩ A^{∗} such that S = F [v_{1}, ⋯, v_{n}] and v_{i_{1}} ⋯ v_{i_{l}} ∈ Alt(A, σ) for 1 ≤ i_{1} < ⋯ < i_{l} ≤ n and 1 ≤ l ≤ n. We call the set {v_{1}, ⋯, v_{n}}, a set of alternating generators of S.

Definition 5.2. Let (A, σ) be a totally decomposable algebra of degree 2^{n} with involution of orthogonal type over a field F of characteristic 2 and let S ⊆ Sym(A, σ) be a ρ-generated totally singular conic algebra such that C_{A}(S) = S. Define a map 𝔰 : S × S → F as follows: for every v, w ∈ S, as vw ∈ S ⊆ Alt(A, σ) + F, there exists a unique α ∈ F such that vw + α ∈ Alt(A, σ). Set 𝔰(v, w) = α. It is easy to see that 𝔰 is a symmetric bilinear form on S. Note that for every v ∈ S, as v^{2} ∈ F and F ∩ Alt(A, σ) = {0}, we obtain

𝔰(v, v) = v^{2} ∈ F. (16)

Furthermore for every u, v, w ∈ S and α ∈ F, we have u(vw) + α ∈ Alt(A, σ) if and only if (uv)w + α ∈ Alt(A, σ), so 𝔰(u, vw) = 𝔰(uv, w), i.e., 𝔰 is an associative bilinear form. We also have the orthogonal decomposition S = F ⊥ S_{0} with respect to 𝔰, where S_{0} = S ∩ Alt(A, σ).

Remark 5.3. Let (A, σ) be a totally decomposable algebra of degree 2^{n} with involution of orthogonal type over a field F of characteristic 2 and let S ⊆ Sym(A, σ) be a ρ-generated totally singular conic algebra such that C_{A}(S) = S. If {v_{1}, ⋯, v_{n}} is a set of alternating generators of S with v_{i}^{2} = α_{i} ∈ F^{×} and 𝔰 is the bilinear form defined in (5.2), then 𝔰 ≃ ⟨⟨α_{1}, ⋯, α_{n}⟩⟩; in particular 𝔰 is nondegenerate.

Definition 5.4. Let (A, σ) ≃ ⨂_{i=1}^{n}(Q_{i}, σ_{i}) be a totally decomposable algebra with involution of orthogonal type over a field F of characteristic 2 and let α_{i} ∈ F^{×}, i = 1, ⋯, n, be a representative of the class disc σ_{i} ∈ F^{×}/F^{×2}. In [7] it is shown that the bilinear n-fold Pfister form 𝔓𝔣(A, σ) := ⟨⟨α_{1}, ⋯, α_{n}⟩⟩ is independent of the decomposition of (A, σ). As in [7] we call this form the Pfister invariant of (A, σ).

The following result gives another description of 𝔓𝔣(A, σ):

Lemma 5.5. Let (A, σ) be a totally decomposable algebra with involution of orthogonal type over a field F of characteristic 2 and let S ⊆ Sym(A, σ) be a ρ-generated totally singular conic algebra such that C_{A}(S) = S. Then the bilinear form 𝔰 on S defined in (5.2) is isometric to 𝔓𝔣(A, σ).

Proof. By (4.3) there exists a σ-invariant quaternion algebra Q_{i} ⊆ A, i = 1, ⋯ , n with (A, σ) ≃ ⨂_{i=1}^{n}(Q_{i}, σ|_{Q_{i}}) and an invertible element v_{i} ∈ Alt(Q_{i}, σ|_{Q_{i}}) such that S = F [v_{1}, ⋯, v_{n}] and v_{i}^{2} = α_{i} ∈ F^{×}, i = 1, ⋯, n. Then {v_{1}, ⋯, v_{n}} is set of alternating generators of S and 𝔰 ≃ ⟨⟨α_{1}, ⋯, α_{n}⟩⟩. We also have disc σ_{i}= Nrd_{Q_{i}}(v_{i})F^{×2} = α_{i}F^{×2} ∈ F^{×}/F^{×2}, so 𝔓𝔣(A, σ) ≃ ⟨⟨α_{1}, ⋯, α_{n}⟩⟩ ≃ 𝔰. □

Let α_{1}, ⋯, α_{n}, β_{1}, ⋯, β_{n} ∈ F^{×}. The bilinear Pfister forms ⟨⟨α_{1}, ⋯, α_{n}⟩⟩ and ⟨⟨β_{1}, ⋯, β_{n}⟩⟩ are said to be simply P-equivalent if either n = 1 and α_{1}F^{×2} = β_{1}F^{×2} or n ≥ 2 and there exist 1 ≤ i < j ≤ n such that ⟨⟨α_{i}, α_{j}⟩⟩ ≃ ⟨⟨β_{i}, β_{j}⟩⟩ and α_{k} = β_{k} for k ≠ i, j. We say that two bilinear Pfister forms 𝔟 and 𝔠 are chain P-equivalent, if there exist bilinear Pfister forms 𝔟_{0}, ⋯, 𝔟_{m} such that 𝔟 = 𝔟_{0}, 𝔠 = 𝔟_{m} and for every i = 0, ⋯, m − 1, 𝔟_{i} and 𝔟_{i+1} are simply P-equivalent.

Proposition 5.6. Let (A, σ) be a totally decomposable algebra of degree 2^{n} with involution of orthogonal type over a field F of characteristic 2 and let S ⊆ Sym(A, σ) be a ρ-generated totally singular conic algebra such that C_{A}(S) = S. Let 𝔰 be the bilinear form on S defined in (5.2). If 𝔰 ≃ ⟨⟨α_{1}, ⋯, α_{n}⟩⟩ for some α_{1}, ⋯, α_{n} ∈ F^{×}, then there exists a set of alternating generators {u_{1}, ⋯, u_{n}} of S such that u_{i}^{2} = α_{i}, i = 1, ⋯, n.

Proof. First suppose that n = 1 and let {v_{1}} is a set of alternating generators of S with v_{1}^{2} = β ∈ F^{×}. Since A is a quaternion algebra, we have disc σ = Nrd_{A}(v_{1}) = βF^{×2} ∈ F^{×}/F^{×2} and the result follows from [15, (7.4)].

Now suppose that n = 2. Set S_{0} = S ∩ Alt(A, σ) = F^{⊥}, 𝔰_{0} = 𝔰|_{S_{0}×S_{0}} and 𝔟 = ⟨⟨α_{1}, α_{2}⟩⟩. As 𝔰 ≃ 𝔟, by [2, p. 16] the pure subforms of these forms are also isometric. So 𝔰_{0} ≃ ⟨α_{1}, α_{2}, α_{1}α_{2}⟩. Let V be the underlying vector space of 𝔟_{0} := ⟨α_{1}, α_{2}, α_{1}α_{2}⟩ with a respective orthogonal basis {v_{1}, v_{2}, v_{3}}. Let f : (V, 𝔟_{0}) ≃ (S_{0}, 𝔰_{0}) be an isometry and set u_{i} = f(v_{i}) ∈ S_{0}, i = 1, 2. We have u_{i}^{2} = f(v_{i})^{2} = v_{i}^{2} = α_{i}, i = 1, 2. We also have 𝔰(1, u_{1}u_{2}) = 𝔰(u_{1}, u_{2}) = 𝔟(v_{1}, v_{2}) = 0, so u_{1}u_{2} ∈ F^{⊥} = S_{0} ⊆ Alt(A, σ). Thus {u_{1}, u_{2}} is the desired set of alternating generators of S.

Finally suppose that n ≥ 3. Let (A, σ) ≃ ⨂_{i=1}^{n}(Q_{i}, σ_{i}) be a decomposition of (A, σ) and let v_{i} ∈ Alt(Q_{i}, σ_{i}) with v_{i}^{2} = β_{i} ∈ F^{×}, i = 1, ⋯, n. Then {v_{1}, ⋯, v_{n}} is set of alternating generators of S and by (5.3) we have 𝔰 ≃ ⟨⟨β_{1}, ⋯, β_{n}⟩⟩, so

⟨⟨α_{1}, ⋯, α_{n}⟩⟩ ≃ ⟨⟨β_{1}, ⋯, β_{n}⟩⟩.

By [2, (A. 1)] there exist bilinear n-fold Pfister forms 𝔟_{0}, ⋯, 𝔟_{m} such that 𝔟_{0} = ⟨⟨β_{1}, ⋯, β_{n}⟩⟩, 𝔟_{m} = ⟨⟨α_{1}, ⋯, α_{n}⟩⟩ and for every i = 0, ⋯, m − 1, 𝔟_{i} is simply Pequivalent to 𝔟_{i+1}. In order to prove the result, using induction on m, it is enough to consider the case where ⟨⟨α_{1}, ⋯, α_{n}⟩⟩ and ⟨⟨β_{1}, ⋯, β_{n}⟩⟩ are simply P-equivalent. By re-indexing if necessary we may assume that ⟨⟨α_{1}, α_{2}⟩⟩ ≃ ⟨⟨β_{1}, β_{2}⟩⟩ and α_{i} = β_{i} for i ≥ 3. Set (B, τ) = (Q_{1}, σ_{1})⊗(Q_{2}, σ_{2}) and S′ = F [v_{1}, v_{2}]. Then S′ ⊆ Sym(B, τ) is a ρ-generated totally singular conic algebra and C_{B}(S′) = S′. As proved in the case n = 2, there exists a set of alternating generators {u_{1}, u_{2}} of S′ such that u_{i}^{2} = α_{i}, i = 1, 2. We have S = S′ ⊗ F [v_{3}, ⋯, v_{n}] ≃ F [u_{1}, u_{2}] ⊗ F [v_{3}, ⋯, v_{n}]. So {u_{1}, u_{2}, v_{3}, ⋯, v_{n}} is the desired set of alternating generators of S. □

Proposition 5.7. Let (A, σ) be a totally decomposable algebra of degree 2^{n} with involution of orthogonal type over a field F of characteristic 2 and let S ⊆ Sym(A, σ) be a ρ-generated totally singular conic algebra such that C_{A}(S) = S. Then the following statements are equivalent: (i) (A, σ) ≃ (M_{2^{n}}(F), t). (ii) 𝔓𝔣(A, σ) ≃ ⟨⟨1, ⋯, 1⟩⟩. (iii) u^{2} ∈ F^{2} for every u ∈ S. (iv) Every maximal subfield of S containing F reduces to F.

Proof. The implication (i) ⇒ (ii) follows from the fact that the transpose involution in characteristic 2 has trivial discriminant, see [15, p. 82].

(ii) ⇒ (i) : Let (A, σ) ≃ ⨂_{i=1}^{n}(Q_{i}, σ_{i}) be a decomposition of (A, σ) to quaternion algebras with involution over F. Since char F = 2 a sum of squares in F is again a square, so D_{F} (𝔓𝔣(A, σ)) = D_{F} (⟨⟨1, ⋯, 1⟩⟩) = F^{2}. It follows that for every i, disc σ_{i} is trivial. So by (4.1) we have (Q_{i}, σ_{i}) ≃ (M_{2}(F), t) which implies that (A, σ) ≃ (M_{2^{n}}(F), t). The equivalence (ii) ⇔ (iii) follows from (5.3), (5.5) and (5.6) and (iii) ⇔ (iv) follows from (3.2). □

The proof of the following result is left to the reader.

Lemma 5.8. Let (A, σ) be a totally decomposable algebra with involution of orthogonal type over a field F of characteristic 2 and let S ⊆ Sym(A, σ) be a ρ-generated totally singular conic F -algebra such that C_{A}(S) = S. If K/F is a field extension, then S_{K} ⊆ Sym(A_{K}, σ_{K}) is a ρ-generated totally singular conic K-algebra and C_{A_{K}} (S_{K}) = S_{K}.

Lemma 5.9. Let (A, σ) be a totally decomposable algebra of degree 2^{n} with involution of orthogonal type over a field F of characteristic 2 and let S ⊆ Sym(A, σ) be a ρ-generated totally singular conic algebra such that C_{A}(S) = S. If K ⊇ F is a maximal subfield of S, then (A_{K}, σ_{K}) ≃ (M_{2^{n}}(K), t). In particular K is a splitting field of A.

Proof. By (5.8), S_{K} ⊆ Sym(A_{K}, σ_{K}) is a ρ-generated totally singular conic algebra and C_{A_{K}} (S_{K}) = S_{K}. As K is a maximal subfield of S, by (3.2) we have u^{2} ∈ K^{2} for every u ∈ S. This, together with K^{2} ⊆ S^{2} ⊆ F implies that x^{2} ∈ K^{2} for every x ∈ S_{K}. So by (5.7) we have (A_{K}, σ_{K}) ≃ (M_{2^{n}}(K), t). □

Let (A, σ) be a totally decomposable algebra with involution of orthogonal type over a field F of characteristic 2. By [7, (7.2)], (5.5) and (5.6) all ρ-generated totally singular conic subalgebras S of (A, σ) with S ⊆ Sym(A, σ) and C_{A}(S) = S are isomorphic as F -algebras. Here, we give a proof of this fact which is independent from [7]:

Lemma 5.10. Let (A, σ) be a totally decomposable algebra of degree 2^{n} with involution of orthogonal type over a field F of characteristic 2 and let S and S′ be two ρ-generated totally singular conic subalgebras of (A, σ) such that S, S′ ⊆ Sym(A, σ), C_{A}(S) = S and C_{A}(S′) = S′.

(i) If K ⊇ F and K′ ⊇ F are respectively maximal subfields of S and S′, then K ≃ K′ as F -algebras.

(ii) We have S ≃ S′ as F -algebras.

Proof. By (5.9) we have (A_{K}, σ_{K}) ≃ (M_{2^{n}}(K), t). Also according to (5.8), S′_{K} is a ρ-generated totally singular conic subalgebra of (A_{K}, σ_{K}) with S′_{K} ⊆ Sym(A_{K}, σ_{K}) and C_{A_{K}} (S′_{K}) = S′_{K}. So we have u^{2} ∈ K^{2} for every u ∈ S′_{K} by (5.7). In particular if K′ = F [v′_{1}, ⋯, v′_{r}], where r = r_{F} (K′) and v′_{1}, ⋯, v′_{r} ∈ K′ ⊆ S′ ⊆ S′_{K}, then v′_{i}^{2} ∈ K^{2}, i = 1, ⋯, r. So for every i there exists v_{i} ∈ K such that v_{i}^{2} = v′_{i}^{2} ∈ K^{2}. By (3.17) the linear map f : K′ → K induced by f(v′_{i}) = v_{i} is an F -algebra homomorphism. As K′ is a field, f is a monomorphism. Similarly there exists an F -algebra monomorphism K ↪ K′. So dim_{F} K = dim_{F} K′ and K ≃ K′ as F -algebras. This proves (i). The statement (ii) follows from (i) and (3.16). □

Notation 5.11. Let (A, σ) be a totally decomposable algebra with involution of orthogonal type over a field F of characteristic 2. In view of (5.10 (ii)) there exists, up to isomorphism, a unique ρ-generated totally singular conic algebra S ⊆ Sym(A, σ) such that C_{A}(S) = S. We denote this algebra by Φ(A, σ). Note that if 𝔰 is the bilinear form on Φ(A, σ) defined in (5.2), then 𝔰 ≃ 𝔓𝔣(A, σ) by (5.5).

Remark 5.12. Let (Q, σ) be a quaternion algebra with involution over a field F of characteristic 2. Then Φ(Q, σ) = F + Alt(Q, σ). In fact by (4.5) we have Φ(Q, σ) ⊆ F + Alt(Q, σ) and dim Φ(Q, σ) = 2. As Alt(Q, σ) is one dimensional, we obtain Φ(Q, σ) = F + Alt(Q, σ).

Corollary 5.13. Let (A, σ) and (B, τ) be two totally decomposable algebras with involution of orthogonal type over a field F of characteristic 2. Then we have Φ(A ⊗ B, σ ⊗ τ) ≃ Φ(A, σ) ⊗ Φ(B, τ).

Proof. (i) Set S = Φ(A, σ) ⊗ Φ(B, τ). Then S ⊆ Sym(A ⊗ B, σ ⊗ τ) is a totally singular conic F -algebra and C_{A⊗B}(S) = S. Also by (3.13), S is a ρ-generated algebra. So by (5.10 (ii)), we have Φ(A ⊗ B, σ ⊗ τ) ≃ S. The part (ii) follows from (5.8) and (5.10 (ii)). □

6. Some characterizing properties of the Pfister invariant

Lemma 6.1. (Compare [5, (2.4)]) Let K/F be a finite extension of fields of characteristic 2 and let (Q, σ) be a quaternion algebra with involution of orthogonal type over K. If u^{2} ∈ F for every u ∈ Φ(Q, σ), then there exists a quaternion F -subalgebra Q_{0} ⊆ Q such that σ(Q_{0}) = Q_{0}, i.e., (Q, σ) ≃_{K} (Q_{0}, σ|_{Q_{0}}) ⊗ (K, id).

Proof. The idea of the proof is similar to the proof of (4.2). Set S = Φ(Q, σ). By (5.12) we have S = K + Alt(Q, σ) = K[u], where 0 ≠ u ∈ Alt(Q, σ). Let δ be the K-derivation of S induced by δ(u) = u. By (2.1 (ii)), δ extends to an inner derivation δ_{ξ} of Q for some ξ ∈ Q. Let η = ξ^{2} ∈ Q. As ξ^{2} + ξ ∈ C_{Q}(S) = S, we obtain (ξ^{2} + ξ)^{2} ∈ F and (ξ + ξ^{2})u = u(ξ + ξ^{2}), thus

η^{2} + η = ξ^{4} + ξ^{2} = (ξ^{2} + ξ)^{2} ∈ F, ηu + uη = ξu + uξ = δ(u) = u.

So the F -algebra generated by u and η is a quaternion algebra. As Alt(Q, σ) = Ku, we have η + σ(η) = αu for some α ∈ K. If α ∈ F, then the quaternion F -algebra Q_{0} generated by u and η is invariant under σ. Otherwise α ≠ 0 and the quaternion F -algebra Q_{0} generated by αu and η is invariant under σ. □

Corollary 6.2. Let K/F be a finite extension of fields of characteristic 2. Let (A, σ) be a totally decomposable algebra with involution of orthogonal type over K. If u^{2} ∈ F for every u ∈ Φ(A, σ), then there exists a central simple F -algebra B ⊆ A such that (A, σ) ≃_{K} (B, σ|_{B}) ⊗ (K, id).

Proof. Let (A, σ) ≃_{K} ⨂_{i=1}^{n}(Q_{i}, σ_{i}) be a decomposition of (A, σ) into quaternion K-algebras with involution and choose an invertible element u_{i} ∈ Alt(Q_{i}, σ_{i}), i = 1, ⋯, n. Then we have Φ(A, σ) ≃ K[u_{1}, ⋯, u_{n}], so u_{i}^{2} ∈ F^{×} for i = 1, ⋯, n. It follows from (5.13) that u^{2} ∈ F for every u ∈ Φ(Q_{i}, σ_{i}). So by (6.1), (A, σ) ≃_{K} ⨂_{i=1}^{n}(Q′_{i}, σ|_{Q′_{i}}) ⊗ (K, id), where Q′_{i} is a quaternion F -subalgebra of Q_{i}. □

Lemma 6.3. Let (A, σ) be a totally decomposable algebra of degree 2^{n} with involution of orthogonal type over a field F of characteristic 2. Consider an element u ∈ Φ(A, σ) with u^{2} ∈ F^{×} \ F^{×2}. Set B = C_{A}(u) and K = F [u].

(i) The pair (B, σ|_{B}) is a totally decomposable algebra with involution of orthogonal type over K and Φ(B, σ|_{B}) ≃ Φ(A, σ) as K-algebras.

(ii) There exists a quaternion algebra Q ⊆ A containing u such that σ(Q) = Q.

Proof. (i) Since σ(u) = u, σ|_{B} is of the first kind. We also have 1 ∉ Alt(A, σ) which implies that 1 ∉ Alt(B, σ|_{B}), so σ|_{B} is of orthogonal type. Set S = Φ(A, σ). By (3.8) we have r_{K}(S) = r_{F} (S) − 1 = n − 1. So dim_{K} S = 2^{r_{K}(S)} and using (3.11), S is a Frobenius ρ-generated K-algebra. As dim_{K} S = deg_{K} B, by (2.1 (i)) we have C_{B}(S) = S. By (4.5) and (5.10 (ii)) (B, σ|_{B}) is totally decomposable and Φ(B, σ|_{B}) ≃ S.

(ii) As u^{2} ∈ F for every u ∈ Φ(B, σ|_{B}) ≃ S, by (6.2) there exists a central simple F -algebra B_{0} ⊆ B such that (B, σ|_{B}) ≃_{K} (B_{0}, σ|_{B_{0}}) ⊗ (K, id). Then Q = C_{A}(B_{0}) ⊆ A is a quaternion algebra containing u which is invariant under σ. □

Lemma 6.4. Let (A, σ) be a totally decomposable algebra with involution of orthogonal type over a field F of characteristic 2. Let {u_{1}, ⋯, u_{n}} be a set of alternating generators of Φ(A, σ) and set α_{i} = u_{i}^{2} ∈ F^{×}, i = 1, ⋯, n. Suppose that α_{n} ∉ F^{2} and set B = C_{A}(u_{n}) and K = F [u_{n}]. Then (B, σ|_{B}) is a totally decomposable algebra with involution of orthogonal type over K and {u_{1}, ⋯, u_{n−1}} is a set of alternating generators of Φ(B, σ|_{B}). In particular 𝔓𝔣(B, σ|_{B}) ≃ ⟨⟨α_{1}, ⋯, α_{n−1}⟩⟩_{K}.

Proof. By (6.3 (i)), (B, σ|_{B}) is a totally decomposable algebra with involution of orthogonal type over K and we have an isomorphism of K-algebras Φ(B, σ|_{B}) ≃ Φ(A, σ). Let 1 ≤ l ≤ n − 1 and 1 ≤ i_{1} < ⋯ < i_{l} ≤ n − 1. We claim that u_{i_{1}} ⋯ u_{i_{l}} ∈ Alt(B, σ|_{B}). By (4.5 (iii)) we have Φ(B, σ|_{B}) ⊆ Alt(B, σ|_{B}) + K, so there exists λ ∈ K such that w := u_{i_{1}} ⋯ u_{i_{l}} + λ ∈ Alt(B, σ|_{B}). Write λ = α + βu_{n} for some α, β ∈ F. Then

w = u_{i_{1}} ⋯ u_{i_{l}} + α + βu_{n} ∈ Alt(B, σ|_{B}) ⊆ Alt(A, σ).

As σ is of orthogonal type, we have α = 0. As u_{n} ∈ K = Z(B) we have u_{n}w ∈ Alt(B, σ|_{B}). On the other hand u_{n}w = u_{n}u_{i_{1}} ⋯ u_{i_{l}} + α_{n}β ∈ Alt(B, σ|_{B}) ⊆ Alt(A, σ). Again, as σ is of orthogonal type, we have β = 0. So u_{i_{1}} ⋯ u_{i_{l}} ∈ Alt(B, σ|_{B}) and the claim is proved, i.e., {u_{1}, ⋯, u_{n−1}} is a set of alternating generators of Φ(B, σ|_{B}). □

In [7] it was shown that if (A, σ) ≃ (A′, σ′) then A ≃ A′ and 𝔓𝔣(A, σ) ≃ 𝔓𝔣(A′, σ′). It was also asked whether the converse is also true (see [7, (7.4)]). The following result shows that this question has an affirmative answer, i.e., totally decomposable algebras with involution of orthogonal type can be classified, up to conjugation, by their Pfister invariant.

Theorem 6.5. Let (A, σ) and (A′, σ′) be two totally decomposable algebras with involution of orthogonal type over a field F of characteristic 2. If A ≃ A′ and 𝔓𝔣(A, σ) ≃ 𝔓𝔣(A′, σ′), then (A, σ) ≃ (A′, σ′).

Proof. Let {u_{1}, ⋯, u_{n}} be a set of alternating generators of Φ(A, σ) with u_{i}^{2} = α_{i} ∈ F^{×}, so that 𝔓𝔣(A, σ) ≃ ⟨⟨α_{1}, ⋯, α_{n}⟩⟩. By (5.5) and (5.6) there exists a set of alternating generators {u′_{1}, ⋯, u′_{n}} of Φ(A′, σ′) such that u′_{i}^{2} = α_{i}, i = 1, ⋯, n.

We use induction on n. If n = 1, we have disc σ = disc σ′ = Nrd_{A}(u_{1})F^{×2} = α_{1}F^{×2}, so the result follows from [15, (7.4)]. So suppose that n > 1. If α_{i} ∈ F^{×2} for every i = 1, ⋯, n, then using (5.7) we obtain (A, σ) ≃ (A′, σ′) ≃ (M_{2^{n}}(F), t) and we are done. So (by re-indexing if necessary) we may assume that α_{n} ∈ F^{×} \ F^{×2}. Set B = C_{A}(u_{n}), K = F [u_{n}], B′ = C_{A′}(u′_{n}) and K′ = F [u′_{n}]. As K ≃ K′ = F (\sqrt{α}_{n}), we may consider B′ as a central simple algebra over K. By (6.3 (i)) and (6.4), (B, σ|_{B}) and (B′, σ′|_{B′}) are totally decomposable algebras with involution of orthogonal type over K and 𝔓𝔣(B, σ|_{B}) ≃ 𝔓𝔣(B′, σ′|_{B′}) ≃ ⟨⟨α_{1}, ⋯, α_{n−1}⟩⟩_{K}. Since A ≃_{F} A′, we obtain B ≃_{K} B′, so by induction hypothesis there exists an isomorphism of K-algebras with involution

f : (B, σ|_{B}) ≃_{K} (B′, σ′|_{B′}).

By (6.3 (i)) we have a K-algebra isomorphism Φ(B, σ|_{B}) ≃ Φ(A, σ). So we get u^{2} ∈ F for every u ∈ Φ(B, σ|_{B}). By (6.2) there exists a central simple F -subalgebra B_{0} ⊆ B such that (B, σ|_{B}) ≃_{K} (B_{0}, σ|_{B_{0}}) ⊗ (K, id). Set B′_{0} = f(B_{0}) ⊆ B′, hence (B_{0}, σ|_{B_{0}}) ≃_{F} (B′_{0}, σ′|_{B′_{0}}). Then B′_{0} is a σ′-invariant central simple subalgebra of B′ and (B′, σ′|_{B′}) ≃_{K} (B′_{0}, σ′|_{B′_{0}}) ⊗ (K, id). Set Q := C_{A}(B_{0}) ⊇ C_{A}(B) = C_{A}(C_{A}(u_{n})), hence u_{n} ∈ Q. Similarly set Q′ := C_{A′}(B′_{0}) = C_{A′}(f(B_{0})) ⊇ C_{A′}(f(B)) = C_{A′}(B′) = C_{A′}(C_{A′}(u′_{n})), hence u′_{n} ∈ Q′. As deg_{F} Q = deg_{F} Q′ = 2, Q and Q′ are quaternion algebras over F. We also have (A, σ) ≃_{F} (Q, σ|_{Q}) ⊗ (B_{0}, σ|_{B_{0}}) and (A′, σ′) ≃_{F} (Q′, σ′|_{Q′}) ⊗ (B′_{0}, σ′|_{B′_{0}}). Since B_{0} ≃_{F} B′_{0} and A ≃_{F} A′, we obtain

Q′ ≃_{F} C_{A′}(B′_{0}) ≃_{F} C_{A}(B_{0}) ≃_{F} Q.

Also as u_{n} ∈ Alt(A, σ) ∩ Q, by [18, (3.5)] we have u_{n} ∈ Alt(Q, σ|_{Q}). It follows that disc σ|_{Q} = Nrd_{Q}(u_{n})F^{×2} = α_{n}F^{×2} and similarly disc σ′|_{Q′} = α_{n}F^{×2}, hence (Q, σ|_{Q}) ≃_{F} (Q′, σ′|_{Q′}) by [15, (7.4)]. Using this isomorphism we obtain

(A, σ) ≃_{F} (Q, σ|_{Q}) ⊗ (B_{0}, σ|_{B_{0}}) ≃_{F} (Q′, σ′|_{Q′}) ⊗ (B′_{0}, σ′|_{B′_{0}}) ≃ (A′, σ′).  □

A bilinear space (V, 𝔟) over a field F is called metabolic if there exists a subspace W of V such that dim W = \frac{1}{2} dim V and 𝔟|_{W ×W} = 0. An F -algebra with involution (A, σ) is called metabolic if there exists an idempotent e ∈ A such that σ(e)e = 0 and dim_{F} eA = \frac{1}{2} dim_{F} A. A bilinear form is metabolic if and only if its adjoint involution is metabolic, see [6, (4.8)].

As an application we complement a characterization of totally decomposable algebras with metabolic involution given in [7].

Theorem 6.6. ([7, (7.5)]) Let (A, σ) be a totally decomposable algebra of degree 2^{n} with involution of orthogonal type over a field F of characteristic 2. Then the following statements are equivalent:

(i) (A, σ) is metabolic.

(ii) 𝔓𝔣(A, σ) is metabolic.

(iii) Φ(A, σ) is not a field.

(iv) There exists a central simple algebra with involution of orthogonal type (B, τ) over F such that (A, σ) ≃ (M_{2}(F), t) ⊗ (B, τ).

Proof. The equivalence of (i) and (ii) was shown in [7, (7.5)].

(ii) ⇒ (iii): If 𝔓𝔣(A, σ) is metabolic, then as 𝔓𝔣(A, σ) ≃ 𝔰, by the relation (16) given in (5.2) there exists a nonzero x ∈ Φ(A, σ) such that x^{2} = 0 and we obtain (iii).

(iii) ⇒ (iv): We use induction on n. If n = 1, the result follows from (5.7) and (5.12). So suppose that n > 1. Let 𝔪 be the unique maximal ideal of Φ(A, σ) and let K ⊇ F be a maximal subfield of Φ(A, σ). If K = F the result again follows from (5.7); so suppose that K ≠ F. Write K = F [u_{1}, ⋯, u_{r}], where r = r_{F} (K) and u_{1}, ⋯, u_{r} ∈ K. Since Φ(A, σ) is not a field we have 𝔪 ≠ {0}. So using (3.9) one can find u_{r+1}, ⋯, u_{n} ∈ 𝔪 such that Φ(A, σ) = K[u_{r+1}, ⋯, u_{n}]. It follows that Φ(A, σ) = F [u_{1}, ⋯, u_{n}] with u_{1} ∈ K \ F and u_{n} ∈ 𝔪. By (4.2 (ii)) there exists a σ-invariant quaternion subalgebra Q of A such that F [u_{2}, ⋯, u_{n}] ⊆ C_{A}(Q). Set B = C_{A}(Q). We have dim_{F} F [u_{2}, ⋯, u_{n}] = deg_{F} B = 2^{n−1}, so by (3.11), F [u_{2}, ⋯, u_{n}] is a ρ-generated F -algebra. Also it is easy to see that C_{B}(F [u_{2}, ⋯, u_{n}]) = F [u_{2}, ⋯, u_{n}]. As F [u_{2}, ⋯, u_{n}] ⊆ Sym(B, σ|_{B}), by (4.5), (B, σ|_{B}) is a totally decomposable algebra with involution of orthogonal type over F. By (5.10 (ii)) we have Φ(B, σ|_{B}) ≃ F [u_{2}, ⋯, u_{n}]. As u_{n} ∈ F [u_{2}, ⋯, u_{n}] ∩ 𝔪, Φ(B, σ|_{B}) is not a field. So the result follows from induction hypothesis.

(iv) ⇒ (i): Let

e = ( \begin{matrix}0 & 1 \\ 0 & 1 \end{matrix} ) ∈ M_{2}(F).

Then e is a metabolic idempotent for (M_{2}(F), t). So (M_{2}(F), t) is metabolic which implies that (A, σ) ≃ (M_{2}(F), t) ⊗ (B, τ) is also metabolic. □

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M. G. Mahmoudi, mmahmoudi@sharif.ir, A.-H. Nokhodkar, anokhodkar@yahoo.com

Department of Mathematical Sciences, Sharif University of Technology, P. O. Box 11155-9415, Tehran, Iran.