POWER-CENTRAL ELEMENTS IN TENSOR PRODUCTS OF SYMBOL ALGEBRAS

DEMBA BARRY

Abstract. Let A be a central simple algebra over a field F. Let k_{1}, …, k_{r} be cyclic extensions of F such that k_{1} ⊗_{F} ⋯ ⊗_{F} k_{r} is a field. We investigate conditions under which A is a tensor product of symbol algebras where each k_{i} is in a symbol F -algebra factor of the same degree as k_{i}. As an application, we give an example of an indecomposable algebra of degree 8 and exponent 2 over a field of 2-cohomological dimension 4.

Keywords Central simple algebra. Symbol algebra, Armature, Valuation, Cohomological dimension

Mathematics Subject Classification (2010) 16K20, 16W60, 12E15, 12G10, 16K50

1. Introduction

Let F be a field and let k be a Galois extension of F of degree n with cyclic group generated by σ. For a ∈ F^{×}, we let (k, σ, a) denote the cyclic F -algebra generated over k by a single element y with defining relation ycy^{−1} = σ(c) for c ∈ k and y^{n} = a. If F contains a primitive n-th root of unity ζ, it follows from Kummer theory that one may write k in the form k = F ( \sqrt[n]{b}) for some b ∈ F^{×}. The algebra (k, σ, a) is then isomorphic to the symbol algebra (a, b)_{n} over F, that is, a central simple F -algebra generated by two elements i and j satisfying i^{n} = a, j^{n} = b and ij = ζji (see for instance [P, §15. 4]). In the case n = 2, ζ = −1, one gets a quaternion algebra over F that will be denoted (a, b).

A central division algebra decomposes into a tensor product of symbol algebras (of degree 2 in [ART] and of degree an arbitrary prime p in [Ro]) if and only if it contains a set whose elements satisfy some commuting properties: q-generating set in [ART], p-central set in [Ro], and a set of representatives of an armature in [T_{1}]. Our approach is based on these notions.

The main goal of this paper is to further investigate the decomposability of central simple algebras; the study of power central-elements is a constant tool. Let A be a central simple algebra over F and let k_{1}, …, k_{r} be cyclic extensions of F, of respective degree n_{1}, …, n_{r}, contained in A such that k_{1} ⊗_{F} ⋯ ⊗_{F} k_{r} is a field. It is natural to ask: when does there exist a decomposition of A into a tensor product of symbol algebras in which each k_{i} is in a symbol F -subalgebra factor of A of degree n_{i}? Starting from A, we construct a division algebra E whose center is the iterated Laurent series with r indeterminates over F and show that A admits such a decomposition if and only if E is a tensor product of symbol algebras (see Theorem 3.1 and Corollary 3.2 for details). Note that Corollary 3.2 is very close to a result of Tignol [T_{2}, Prop. 2.10]. In contrast with Tignol’s result (which is stated in terms of Brauer equivalence and only for prime exponent), Corollary 3.2 is stated in terms of isomorphism classes and is valid for any exponent. Moreover, our approach is completely different. We will give an example, pointed out by Merkurjev, of a division algebra which is a tensor product of three quaternion algebras, and containing a quadratic field extension which is in no quaternion subalgebra (Corollary 4.6). Using valuation theory, we give a general method for constructing tensor products of quaternion algebras containing a quadratic field extension which is in no quaternion subalgebra. As an application, let A be a central simple algebra of degree 8 and exponent 2 over F, and containing a quadratic field extension which is in no quaternion subalgebra. We use Corollary 3.2 to associate with A an example of an indecomposable algebra of degree 8 and exponent 2 over a field of rational functions in one variable over a field of 2-cohomological dimension 3 (see Theorem 4.8). This latter field is obtained by an inductive process pioneered by Merkurjev [M].

We next recall some results related to our main question: let A be a 2-power dimensional central simple algebra over F and let F (\sqrt{d_{1}}, \sqrt{d_{2}}) ⊂ A be a biquadratic field extension of F. If A is a biquaternion algebra, it follows from a result of Albert that A ≃ (d_{1}, d′_{1}) ⊗_{F} (d_{2}, d′_{2}) where d′_{1}, d′_{2} ∈ F^{×} (see for instance [Ra]). As observed above, this is not true anymore in higher degree. More generally, if A is of degree 8 and exponent 2 and F (\sqrt{d}) ⊂ A is a quadratic field extension, there exists a cohomological criterion associated with the centralizer of F (\sqrt{d}) in A which determines whether F (\sqrt{d}) lies in a quaternion F -subalgebra of A (see [Ba, Prop. 4.4]). In the particular case where the 2-cohomological dimension of F is 2 and A is a division algebra of exponent 2 over F, the situation is more favorable: it is shown in [Ba, Thm. 3.3] that there exits a decomposition of A into a tensor product of quaternion F -algebras in which each F (\sqrt{d_{i}}) (for i = 1, 2) is in a quaternion F -subalgebra.

An outline of this article is the following: in Section 2 we collect from [T_{1}] and [TW] some results on armatures of algebras that will be used in the proofs of the main results. Section 3 is devoted to the statements and the proofs of the main results. The particular case of exponent 2 is analyzed (in more details) in Section 4.

All algebras considered in this paper are associative and finite-dimensional over their center. A central simple algebra A over a field F is decomposable if A ≃ A_{1} ⊗_{F} A_{2} for two central simple F -algebras A_{1} and A_{2} both non isomorphic to F; otherwise A is called indecomposable.

Throughout this article, we shall use freely the standard terminology and notation from the theory of finite-dimensional algebras and the theory of valuations on division algebras. For these, as well as background information, we refer the reader to Pierce’s book [P].

2. Armatures of algebras

Armatures in central simple algebras are a major tool for the next section. The goal of this section is to recall the notion of an armature and gather some preliminary results that will be used in the sequel.

We write |H| for the cardinality of a set H. Let A be a central simple F -algebra. For a ∈ A^{×}/F^{×}, we fix an element x_{a} of A whose image in A^{×}/F^{×} is a, that is, a = x_{a}F^{×}. For a finite subgroup A of A^{×}/F^{×},

F [A] = { ∑_{a∈A} c_{a}x_{a} | c_{a} ∈ F }

denotes the F -subspace of A generated by {x_{a} | a ∈ A}. Note that this subspace is independent of the choice of representatives x_{a} for a ∈ A. Since A is a group, F [A] is the subalgebra of A generated by {x_{a} | a ∈ A}. As was observed in [T_{1}], if A is a finite abelian subgroup of A^{×}/F^{×} there is an associated pairing ⟨ , ⟩ on A × A defined by

⟨a, b⟩ = x_{a}x_{b}x_{a}^{−1}x_{b}^{−1}.

This definition is independent of the choice of representatives x_{a}, x_{b} for a, b and ⟨a, b⟩ belongs to F^{×} as A is abelian. Hence, ⟨a, b⟩ is central in A, and it follows that the pairing ⟨ , ⟩ is bimultiplicative. It is also alternating, obviously. Thus, as A is finite, the image of ⟨ , ⟩ is a finite subset of μ(F) (where μ(F) denotes the group of roots of unity of F). For any subgroup H of A let

H^{⊥} = {a ∈ A | ⟨a, h⟩ = 1 for all h ∈ H},

a subgroup of A. The subgroup H^{⊥} is called the orthogonal of H with respect to ⟨ , ⟩. The radical of A, rad(A), is defined to be A^{⊥}. The pairing ⟨ , ⟩ is called nondegenerate on A if rad(A) = {1_{A}}.

For g ∈ A, we denote by (g) the cyclic subgroup of A generated by g . The set {g_{1}, …, g_{r}} is called a base of A if A is the internal direct product

A = (g_{1}) × ⋯ × (g_{r}).

If ⟨ , ⟩ is nondegenerate then A has a symplectic base with respect to ⟨ , ⟩, i.e, a base {g_{1}, h_{1}, …, g_{n}, h_{n}} such that for all i, j

⟨g_{i}, h_{i}⟩ = c_{i}, where ord(g_{i}) = ord(h_{i}) = ord(c_{i})
⟨g_{i}, g_{j}⟩ = ⟨h_{i}, h_{j}⟩ = 1 and, if i ≠ j, ⟨g_{i}, h_{j}⟩ = 1

(see [T_{1}, 1.8]).

Definition 2.1. For any finite-dimensional F -algebra A, a subgroup A of A^{×}/F^{×} is an armature of A if A is abelian, |A| = dim_{F} A, and F [A] = A.

If A = {a_{1}, …, a_{n}} is an armature of A, the above definition shows that the set {x_{a_{1}}, …, x_{a_{n}}} is an F -base of A. The notion of an armature was introduced by Tignol in [T_{1}] for division algebras. The definition given here, slightly different from that given in [T_{1}], comes from [TW]. This definition allows armatures in algebras other than division algebras. The following examples will be used repeatedly in the next section.

Examples 2.2. (a) ([T_{1}]) Let A = A_{1} ⊗_{F} ⋯ ⊗_{F} A_{r} be a tensor product of symbol F -algebras where A_{k} is a symbol subalgebra of degree n_{k}. Suppose F contains a primitive n_{k}-th root ζ_{k} of unity for k = 1, …, r. So, A_{k} is isomorphic to a symbol algebra (a_{k}, b_{k})_{ζ_{k}} of degree n_{k} for some a_{k}, b_{k} ∈ F^{×}. For each k, let i_{k}, j_{k} be a symbol generator of (a_{k}, b_{k})_{ζ_{k}}. The image A in A^{×}/F^{×} of the set

{i_{1}^{α_{1}}j_{1}^{β_{1}} … i_{r}^{α_{r}}j_{r}^{β_{r}} | 0 ≤ α_{k}, β_{k} ≤ n_{k} − 1}

is an armature of A isomorphic to (ℤ/n_{1}ℤ)^{2} × ⋯ × (ℤ/n_{r}ℤ)^{2}. We observe that for all 1 ≠ a ∈ A there exists b ∈ A such that ⟨a, b⟩ ≠ 1; that is the pairing ⟨ , ⟩ is nondegenerate on A. Furthermore, {i_{1}F^{×}, j_{1}F^{×}, …, i_{r}F^{×}, j_{r}F^{×}} is a symplectic base of A.

(b) Let M be a finite abelian extension of a field F and let G be the Galois group of M over F. Let ℓ be the exponent of G. If F contains an ℓ-th primitive root of unity, the extension M/F is called a Kummer extension. Let

S = {x ∈ M^{×} | x^{ℓ} ∈ F^{×}} and Kum(M/F) = S/F^{×}.

It follows from Kummer theory (see for instance [J_{1}, p.119-123]) that Kum(M/F) is a subgroup of M^{×}/F^{×} and is dual to G by the nondegenerate Kummer pairing G × Kum(M/F) → μ(F) given by (σ, b) = σ(x_{b})x_{b}^{−1}, for σ ∈ G and b ∈ Kum(M/F). Whence, Kum(M/F) is isomorphic (not canonically in general) to G. As observed in [TW, Ex. 2.4], the subgroup Kum(M/F) is the only armature of M with exponent dividing ℓ.

Let A be an armature of a central simple F -algebra A and let {a_{1}, b_{1}, …, a_{n}, b_{n}} be a symplectic base of A with respect to ⟨ , ⟩. We shall denote by F [(a_{k})×(b_{k})] the subalgebra of A generated by the representatives x_{a_{k}} and x_{b_{k}} of a_{k} and b_{k}. It is clear that F [(a_{k}) × (b_{k})] is a symbol subalgebra of A of degree n_{k} = ord(a_{k}) = ord(b_{k}) and generated by x_{a_{k}} and x_{b_{k}}. It is shown in [TW, Lemma 2.5] that A ≃ F [(a_{1}) × (b_{1})] ⊗_{F} ⋯ ⊗_{F} F [(a_{n}) × (b_{n})].

Actually, the notion of an armature is a generalization and refinement of the notion of a quaternion generating set (q-generating set) introduced in [ART]. Indeed, a central simple algebra A over F has an armature if and only if A is isomorphic to a tensor product of symbol algebras over F (see [TW, Prop. 2.7]).

Note that if the exponent of A is a prime p, we may consider A as a vector space over the field with p elements 𝔽_{p}. Identifying the group of p-th roots of unity with 𝔽_{p} by a choice of a primitive p-th root of unity, we may suppose the pairing has values in 𝔽_{p}. So, two elements a, b ∈ A are orthogonal if and only if ⟨a, b⟩ = 0. We need the following proposition:

Proposition 2.3. Let V be a vector space over 𝔽_{p} of dimension 2n and let ⟨ , ⟩ be a nondegenerate alternating pairing on V. Let {e_{1}, …, e_{r}} be a base of a totally isotropic subspace of V with respect to ⟨ , ⟩. There are f_{1}, …, f_{r}, e_{r+1}, f_{r+1}, …, e_{n}, f_{n} in V such that {e_{1}, f_{1}, …, e_{n}, f_{n}} is a symplectic base of V.

Proof. We argue by induction on the dimension of the totally isotropic subspace spanned by e_{1}, …, e_{r}. If r = 1, since the pairing is nondegenerate, there is f_{1} ∈ V such that ⟨e_{1}, f_{1}⟩ ≠ 0 . Denote by U = span(e_{1}, f_{1}) the subspace spanned by e_{1}, f_{1}. We have V = U ⊥ U^{⊥} since the restriction of ⟨ , ⟩ to U is nondegenerate. We take for {e_{2}, f_{2}, …, e_{n}, f_{n}} a symplectic base of U^{⊥}.

Assume the statement for a totally isotropic subspace of dimension r − 1. Let W = span(e_{2}, …, e_{r}); we have W ⊂ W^{⊥}. First, we find f_{1} ∈ W^{⊥} such that ⟨e_{1}, f_{1}⟩ ≠ 0. For this, consider the induced pairing, also denoted by ⟨ , ⟩, on W^{⊥}/W defined by ⟨x + W, y + W ⟩ = ⟨x, y⟩ for x, y ∈ W^{⊥}. It is well-defined, and nondegenerate since (W^{⊥})^{⊥} = W. The element e_{1} + W being non-zero in W^{⊥}/W, there is f_{1} + W ∈ W^{⊥}/W such that 0 ≠ ⟨e_{1}, f_{1}⟩ = ⟨e_{1} + W, f_{1} + W ⟩. Letting U = span(e_{1}, f_{1}), we have V = U ⊥ U^{⊥} and e_{2}, …, e_{r} ∈ U^{⊥}. Induction yields f_{2}, …, f_{r}, e_{r+1}, f_{r+1}, …, e_{n}, f_{n} ∈ U^{⊥} such that {e_{2}, f_{2}, …, e_{n}, f_{n}} is a symplectic base of U^{⊥}. Then {e_{1}, f_{1}, …, e_{n}, f_{n}} is a symplectic base of V. □

3. Decomposability

Let A be a central simple algebra over F and let t_{1}, …, t_{r} be independent indeterminates over F. For i = 1, …, r, let k_{i} be a cyclic extension of F of degree n_{i} contained in A. We assume that M = k_{1} ⊗_{F} ⋯ ⊗_{F} k_{r} is a field and denote by G the Galois group of M over F. So [M : F] = n_{1} … n_{r} and G = ⟨σ_{1}⟩ × ⋯ × ⟨σ_{r}⟩, where ⟨σ_{i}⟩ is the Galois group of k_{i} over F, and the order of G is n_{1} … n_{r}. Every element σ ∈ G can be expressed as σ = σ_{1}^{m_{1}} … σ_{r}^{m_{r}} (0 ≤ m_{i} < n_{i}). We shall denote by C = C_{A}M the centralizer of M in A. Let t_{1}, …, t_{r} be independent indeterminates over F. Consider the fields

L′ = F (t_{1}, …, t_{r}) and L = F ((t_{1})) … ((t_{r}))

and the following central simple algebras over L′ and L respectively

N′ = (k_{1} ⊗_{F} L′, σ_{1} ⊗ id, t_{1}) ⊗_{L′} ⋯ ⊗_{L′} (k_{r} ⊗_{F} L′, σ_{r} ⊗ id, t_{r})

and

N = (k_{1} ⊗_{F} L, σ_{1} ⊗ id, t_{1}) ⊗_{L} ⋯ ⊗_{L} (k_{r} ⊗_{F} L, σ_{r} ⊗ id, t_{r}).

We let

R′ = A ⊗_{F} N′ and R = A ⊗_{F} N.

In this section our goal is to prove the following results:

Theorem 3.1. Let M be a finite group with a nondegenerate alternating pairing M×M → μ(F). Suppose C is a division subalgebra of A, and F contains a primitive exp(M)-th root of unity. Then, the following are equivalent:

(i) The division algebra Brauer equivalent to R′ (respectively R) has an armature isomorphic to M.

(ii) The algebra A has an armature isomorphic to M and containing Kum(M/F) as a totally isotropic subgroup.

In the particular case where A is of degree p^{n} and exponent p (for a prime number p) and each k_{i} is a cyclic extension of F of degree p, we have:

Corollary 3.2. Assume that A is of degree p^{n} and exponent p. Suppose C is a division subalgebra of A, F contains a primitive p-th root of unity, and n_{i} = p for i = 1, …, r. Then, the following are equivalent:

(i) The division algebra Brauer equivalent to R′ (respectively R) is decomposable into a tensor product of symbol algebras of degree p.

(ii) The algebra A decomposes as

A ≃ (k_{1}, σ_{1}, δ_{1}) ⊗_{F} ⋯ ⊗_{F} (k_{r}, σ_{r}, δ_{r}) ⊗_{F} A_{r+1} ⊗_{F} ⋯ ⊗_{F} A_{n}

for some δ_{1}, …, δ_{r} ∈ F^{×} and some symbol algebras A_{r+1}, …, A_{n} of degree p.

Moreover, if these conditions are satisfied then the division algebras Brauer equivalent to R′ and R decompose respectively as

(k_{1} ⊗_{F} L′, σ_{1} ⊗ id, δ_{1}t_{1}) ⊗_{L′} ⋯ ⊗_{L′} (k_{r} ⊗_{F} L′, σ_{r} ⊗ id, δ_{r}t_{r}) ⊗_{F} A_{r+1} ⊗_{F} ⋯ ⊗_{F} A_{n}

and

(k_{1} ⊗_{F} L, σ_{1} ⊗ id, δ_{1}t_{1}) ⊗_{L} ⋯ ⊗_{L} (k_{r} ⊗_{F} L, σ_{r} ⊗ id, δ_{r}t_{r}) ⊗_{F} A_{r+1} ⊗_{F} ⋯ ⊗_{F} A_{n}.

As opposed to part (i), it is not enough in part (ii) of Theorem 3.1 to assume simply that A has an armature to get an armature in the division algebra Brauer equivalent to R′ or R. The following example shows that the existence of an armature and the existence of an armature containing Kum(M/F) are different.

Example 3.3. Denote by ℚ_{2} the field of 2-adic numbers, and let A be a division algebra of degree 4 and exponent 4 over F = ℚ_{2}(\sqrt{−1}). Such an algebra is a symbol (see [P, Th., p. 338]). Note that M = F (\sqrt{2}, \sqrt{5}) is a field since the set {−1, 2, 5} forms a ℤ/2ℤ-basis of ℚ_{2}^{×}/ℚ_{2}^{×2} (see for instance [L, Lemma 2.24, p. 163]). It also follows by [P, Prop., p. 339] that A ⊗ M is split. Hence, since [M : F] divides deg(A), we deduce that M ⊂ A. Moreover, it is clear that the centralizer of M in A is M. The algebra A being a symbol, it has an armature A isomorphic to (ℤ/4ℤ)^{2} (see Example 2.2). Assume that A contains Kum(M/F) ≃ (ℤ/2ℤ)^{2}. Since A^{2} = {a^{2} | a ∈ A} = Kum(M/F), the algebra A is a symbol of the form A = (c, d)_{4} with c.F^{×2}, d.F^{×2} ∈ {2.F^{×2}, 5.F^{×2}, 2.5.F^{×2}}. It follows that A ⊗ A is either Brauer equivalent to the quaternion algebra (2, 5) or (2, 2.5) or (5, 2.5). Since (2, 5) ≃ (−1, −1) over ℚ_{2}, the algebra (2, 5) is split over F; that is A ⊗ A is split. Therefore the exponent of A must be 2; impossible. Therefore A has no armature containing Kum(M/F).

Now, let E be the division algebra Brauer equivalent to A ⊗ (2, t_{1}) ⊗ (5, t_{2}). As we will see soon (Lemma 3.4 and Lemma 3.5), deg(E) = exp(E) = 4. But E has no armature, that is, E is not a symbol. Indeed, suppose E has an armature B. If B ≃ (ℤ/2ℤ)^{4} then E is a biquaternion algebra, so exp(E) = 2; contradiction. Therefore B is isomorphic to (ℤ/4ℤ)^{2}. It follows then by Theorem 3.1 that A has an armature containing the amature Kum(M/F) of M. This is impossible as we showed above; therefore E has no armature.

3.1. Brauer classes of R′ and R. For the proof of the results above, we need an explicit description of the division algebras Brauer equivalent to R′ and R. First, we fix some notation: recall that we denoted by G = ⟨σ_{1}⟩ × ⋯ × ⟨σ_{r}⟩ the Galois group of M over F. By the Skolem-Noether Theorem, for each σ_{i} there exists z_{i} ∈ A^{×} such that σ_{i}(b) = z_{i}bz_{i}^{−1} for all b ∈ M. Notice that z_{i}^{n_{i}} =: c_{i} ∈ C and z_{i}z_{j}z_{i}^{−1}z_{j}^{−1} =: u_{ij} ∈ C for all i, j. For all σ = σ_{1}^{m_{1}} … σ_{r}^{m_{r}} ∈ G, we set z_{σ} = z_{1}^{m_{1}} … z_{r}^{m_{r}} (0 ≤ m_{i} < n_{i}). Setting c(σ, τ) = z_{σ}z_{τ} (z_{στ})^{−1}, a simple observation shows for all σ, τ

(3.1)  c(σ, τ) ∈ C and z_{σ}b = σ(b)z_{σ} for all b ∈ M.

In fact, c(σ, τ) can be calculated from the elements u_{ij} and c_{i}.

Let y_{i} be a generator of (k_{i} ⊗ L′, σ_{i} ⊗ 1, t_{i}), that is an element satisfying the relations y_{i}(d ⊗ l)yi^{−1} = σ_{i}(d) ⊗ l for all d ∈ k_{i} and l ∈ L′ and y_{i}^{n_{i}} = t_{i}. For σ = σ_{1}^{m_{1}} … σ_{r}^{m_{r}} ∈ G with 0 ≤ m_{i} < n_{i}, we set y_{σ} = y_{1}^{m_{1}} … y_{r}^{m_{r}}. The algebras N′ and N being crossed products, we may write

N′ = ⨁_{σ∈G}(M ⊗ L′)y_{σ} and N = ⨁_{σ∈G}(M ⊗ L)y_{σ}

and the y_{σ} satisfy

(3.2)  y_{σ}y_{τ} (y_{στ})^{−1} ∈ M ⊗ L and y_{σ}(b ⊗ l) = (σ(b) ⊗ l)y_{σ}

for all b ∈ M and l ∈ L. Set f(σ, τ) = y_{σ}y_{τ} (y_{στ})^{−1}. The elements f(σ, τ) are in fact in L′. Indeed, let y_{σ} = y_{1}^{α_{1}} … y_{r}^{α_{r}} and y_{τ} = y_{1}^{β_{1}} … y_{r}^{β_{r}} with 0 ≤ α_{i}, β_{i} < n_{i} . We get

(3.3)  f(σ, τ) = y_{σ}y_{τ} (y_{στ})^{−1} = t_{1}^{ε_{1}} … t_{r}^{ε_{r}}

where ε_{i} = 0 if α_{i} + β_{i} < ord(σ_{i}) = n_{i} and ε_{i} = 1 if α_{i} + β_{i} ≥ ord(σ_{i}). Note that f(σ, τ) = f(τ, σ) for all σ, τ ∈ G.

Now, let e be the separability idempotent of M, that is, the idempotent e ∈ M ⊗_{F} M determined uniquely by the conditions that

(3.4)  e · (x ⊗ 1) = e · (1 ⊗ x) for all x ∈ M

and the multiplication map M ⊗_{F} M → M carries e to 1 (see for instance [P, §14. 3]). For σ ∈ G, let

e_{σ} = (id ⊗ σ)(e) ∈ M ⊗_{F} M.

The elements (e_{σ})_{σ∈G} form a family of orthogonal primitive idempotents of M ⊗_{F} M (see [P, §14. 3] ) and it follows, by applying id ⊗ σ to each side of (3.4), that

(3.5)  e_{σ} · (x ⊗ 1) = e_{σ} · (1 ⊗ σ(x)) for x ∈ M.

We need the following lemma:

Lemma 3.4. Notations are as above.

(1) The elements z_{σ} ⊗ y_{σ} are subject to the following rules (i) For all σ, τ ∈ G, there exists u ∈ C_{L} such that

(z_{σ} ⊗ y_{σ})(z_{τ} ⊗ y_{τ}) = (u ⊗ 1)(z_{στ} ⊗ y_{στ}).

(ii) (z_{σ} ⊗ y_{σ})(c ⊗ 1) = ((z_{σ}cz_{σ}^{−1}) ⊗ 1)(z_{σ} ⊗ y_{σ}) for all c ∈ C. Moreover,

z_{σ}cz_{σ}^{−1} ∈ C for all σ ∈ G.

(2) The sums

E′ = ∑_{σ∈G} C_{L′}z_{σ} ⊗ y_{σ} and E = ∑_{σ∈G} C_{L}z_{σ} ⊗ y_{σ}

are direct and are central simple subalgebras of R′ and R respectively. Moreover, the algebras R′ and R are Brauer equivalent to E′ and E respectively, and deg E′ = deg E = deg A.

Proof. (1) The statement of (i) follows from relations (3.1), (3.2) and the fact that f(σ, τ) ∈ L′. Since the elements 1 ⊗ y_{σ} centralize C_{L} ⊗ 1, the statement of (ii) is clear.

(2) Suppose ∑_{σ∈G} c_{σ}z_{σ} ⊗ y_{σ} = 0 with c_{σ} ∈ C_{L′} (or C_{L}). Pick such a sum with a minimal number of non-zero terms. There are at least two non-zero elements, say c_{ρ}z_{ρ} ⊗ y_{ρ}, c_{τ} z_{τ} ⊗ y_{τ}, in the sum. Let b ∈ M be such that ρ(b) ≠ b and τ(b) = b. One has

(b ⊗ 1)( ∑_{σ∈G} c_{σ}z_{σ} ⊗ y_{σ})(b ⊗ 1)^{−1} − ∑_{σ∈G} c_{σ}z_{σ} ⊗ y_{σ} = 0

and the number of non-zero terms is nontrivial and strictly smaller; contradiction. Whence we have the direct sums E′ = ⨁_{σ∈G} C_{L′}z_{σ} ⊗ y_{σ} and E = ⨁_{σ∈G} C_{L}z_{σ} ⊗ y_{σ}. It is clear that E′ ⊂ R′ and E ⊂ R. On the other hand, the computation rules of the part (1) show that E′ and E are generalized crossed products (see [A, Th. 11.11] or [J_{2}, §1.4]). Hence, the same arguments as for the usual crossed products show that E′ and E are central simple algebras over L′ and L respectively. Moreover, since dim C = \frac{1}{n_{1}...n_{r}} dim A, we have deg E′ = deg E = deg A.

Now, it remains to show that R′ and R are respectively Brauer equivalent to E′ and E. For this we work over L; the same arguments apply over L′. For z_{σ} ⊗ y_{σ} as above, consider the inner automorphism

Int(z_{σ} ⊗ y_{σ}) : R ⟶ R.

Notice that Int(z_{σ} ⊗ y_{σ})(e_{τ}) is in M ⊗ M and is a primitive idempotent for all τ ∈ G (since e_{τ} is primitive). Moreover, for x ∈ M,

Int(z_{σ} ⊗ y_{σ})(e_{τ}) · (x ⊗ 1) = (z_{σ} ⊗ y_{σ})e_{τ} (z_{σ}^{−1} ⊗ y_{σ}^{−1}) · (x ⊗ 1)
= (z_{σ} ⊗ y_{σ})e_{τ} · (σ^{−1}(x) ⊗ 1)(z_{σ}^{−1} ⊗ y_{σ}^{−1})
= (z_{σ} ⊗ y_{σ})e_{τ} · (1 ⊗ τσ^{−1}(x))(z_{σ}^{−1} ⊗ y_{σ}^{−1})
= (1 ⊗ τ(x)) · Int(z_{σ} ⊗ y_{σ})(e_{τ}).

Therefore Int(z_{σ}⊗y_{σ})(e_{τ}) = e_{τ} by comparing with the definition of e and the relation (3.5). Hence, each e_{τ} centralizes E in R. On the other hand, since the degree of the centralizer C_{R}E of E in R is n_{1} … n_{r} and (e_{τ})_{τ∈G} ⊂ C_{R}E, the algebra C_{R}E is split. So R is Brauer equivalent to E. □

Lemma 3.5. Notations are as in Lemma 3.4. If C is a division algebra then there exists a unique valuation on E which extends the (t_{1}, …, t_{r})-adic valuation on L. Consequently E′ and E are division algebras.

Proof. The (t_{1}, …, t_{r})-adic valuation on L being Henselian, it extends to a unique valuation to each division algebra over L (see for instance [W_{1}]). It follows that there is a valuation on C_{L} extending the (t_{1}, …, t_{r})-adic valuation on L. More precisely this valuation is constructed as follows: writing C_{L} as

C_{L} = \{ ∑_{i_{1}∈ℤ} ⋯ ∑_{i_{r}∈ℤ} c_{i_{1}...i_{r}}t_{1}^{i_{1}} … t_{r}^{i_{r}} | \begin{matrix}c_{i_{1}...i_{r}} ∈ C and \\ \{(i_{1}, …, i_{r}) | c_{i_{1}...i_{r}} ≠ 0 is well-ordered \\ for the right-to-left lexicographic ordering\} \end{matrix} \},

computations show that the map v : C_{L}^{×} → ℤ^{r} defined by

v( ∑_{i_{1}∈ℤ} ⋯ ∑_{i_{r}∈ℤ} c_{i_{1}...i_{r}}t_{1}^{i_{1}} … t_{r}^{i_{r}}) = min{(i_{1}, …, i_{r}) | c_{i_{1}...i_{r}} ≠ 0}

is a valuation. Clearly v extends the (t_{1}, …, t_{r})-adic valuation on L. Recall that N is a division algebra over L (see e.g. [W_{2}, Ex. 3.6]). We also denote by v the unique extension of the (t_{1}, …, t_{r})-valuation to N. Since y_{i}^{n_{i}} = t_{i}, we have

v(y_{i}) = (0, …, 0, \frac{1}{n_{i}} , 0, …, 0) ∈ \frac{1}{n_{1}} ℤ × … × \frac{1}{n_{r}} ℤ.

Hence, for y_{σ} = y_{1}^{m_{i}} … y_{r}^{m_{r}} where σ = σ_{1}^{m_{1}} … σ_{r}^{m_{r}} (with 1 ≤ m_{i} < n_{i}), we have v(y_{σ}) = (\frac{m_{1}}{n_{1}} , …, \frac{m_{r}}{n_{r}}). It then follows that

(3.6)  v(y_{σ}) ≢ v(y_{τ}) mod ℤ^{r} if σ ≠ τ.

Now, define a map w : E^{×} → \frac{1}{n_{1}} ℤ × … × \frac{1}{n_{r}} ℤ as follows: for any σ ∈ G and any c ∈ C_{L}^{×}, set

w(cz_{σ} ⊗ y_{σ}) = v(c) + v(y_{σ}).

For any s ∈ E^{×}, s has a unique representation s = ∑_{σ∈G} c_{σ}z_{σ} ⊗y_{σ} with the c_{σ} ∈ C_{L} and some c_{σ} ≠ 0. Define

w(s) = min_{σ∈G}{w(c_{σ}z_{σ} ⊗ y_{σ}) | c_{σ} ≠ 0}.

It follows by (3.6) that w(c_{σ}z_{σ} ⊗ y_{σ}) ≠ w(c_{τ} z_{τ} ⊗ y_{τ}) for σ ≠ τ. Thus, there is a unique summand c_{ι}z_{ι} ⊗ y_{ι} of s such that w(s) = w(c_{ι}z_{ι} ⊗ y_{ι}); this c_{ι}z_{ι} ⊗ y_{ι} is called the leading term of s.

We are going to show that w is a valuation on E. Let s′ = ∑_{σ∈G} d_{σ}z_{σ} ⊗ y_{σ} ∈ E^{×} with d_{σ} ∈ C_{L} and s + s′ ≠ 0. Let (c_{ρ} + d_{ρ})z_{ρ} ⊗ y_{ρ} be the leading term of s + s′. If c_{ρ} ≠ 0 and d_{ρ} ≠ 0, we have

w((c_{ρ} + d_{ρ})z_{ρ} ⊗ y_{ρ}) = v(c_{ρ} + d_{ρ}) + v(y_{ρ})
≥ min(v(c_{ρ}) + v(y_{ρ}), v(d_{ρ}) + v(y_{ρ}))
= min(w(c_{ρ}z_{ρ} ⊗ y_{ρ}), w(d_{ρ}z_{ρ} ⊗ y_{ρ}))
≥ min(w(s), w(s′)).

Thus, w(s+s′) = w((c_{ρ} +d_{ρ})z_{ρ} ⊗y_{ρ})) ≥ min(w(s), w(s′)). This inequality still holds if c_{ρ} = 0 or d_{ρ} = 0.

By the usual argument, we also check that

(3.7)  if w(s) ≠ w(s′) then w(s + s′) = min(w(s), w(s′)).

It remains to show that w(ss′) = w(s) + w(s′). For σ, τ ∈ G, recall that

(z_{σ} ⊗ y_{σ})(z_{τ} ⊗ y_{τ}) = c(σ, τ)f(σ, τ)(z_{στ} ⊗ y_{στ})

for some c(σ, τ) ∈ C^{×} and some f(σ, τ) ∈ L^{×}. It follows that, for c_{σ}, d_{τ} ∈ C_{L}^{×},

w((c_{σ}z_{σ} ⊗ y_{σ})(d_{τ} z_{τ} ⊗ y_{τ})) = w(c_{σ}(z_{σ}d_{τ} z_{σ}^{−1})(z_{σ} ⊗ y_{σ})(z_{τ} ⊗ y_{τ})) (Lemma 3.4)
= w(c_{σ}(z_{σ}d_{τ} z_{σ}^{−1})c(σ, τ)f(σ, τ)(z_{στ} ⊗ y_{στ}))
= v(c_{σ}) + v(d_{τ}) + v(f(σ, τ)y_{στ})
= v(c_{σ}) + v(y_{σ}) + v(d_{τ}) + v(y_{τ})
(3.8)  = v(c_{σ}z_{σ} ⊗ y_{σ}) + v(d_{τ} z_{τ} ⊗ y_{τ}).

Hence, we have

w(ss′) = w( ∑_{σ,τ}(c_{σ}z_{σ} ⊗ y_{σ})(d_{τ} z_{τ} ⊗ y_{τ}))
≥ min_{σ,τ}{w((c_{σ}z_{σ} ⊗ y_{σ})(d_{τ} z_{τ} ⊗ y_{τ})) | c_{σ}, d_{τ} ≠ 0}
= min_{σ,τ}{w(c_{σ}z_{σ} ⊗ y_{σ}) + w(d_{τ} z_{τ} ⊗ y_{τ}) | c_{σ}, d_{τ} ≠ 0}
(3.9)  ≥ w(s) + w(s′).

Let c_{ρ}z_{ρ} ⊗ y_{ρ} and d_{ι}z_{ι} ⊗ y_{ι} be the leading terms of s and s′ respectively. Set s_{1} = s − c_{ρ}z_{ρ} ⊗ y_{ρ} and s′_{1} = s′ − d_{ι}z_{ι} ⊗ y_{ι}. So,

w(s) = w(c_{ρ}z_{ρ} ⊗ y_{ρ}) < w(s_{1}) and w(s′) = w(d_{ι}z_{ι} ⊗ y_{ι}) < w(s′_{1}).

Writing

ss′ = (c_{ρ}z_{ρ} ⊗ y_{ρ})(d_{ι}z_{ι} ⊗ y_{ι}) + s_{1}(d_{ι}z_{ι} ⊗ y_{ι}) + (c_{ρ}z_{ρ} ⊗ y_{ρ})s′_{1} + s_{1}s′_{1},

it follows by (3.8) and (3.9) that the first summand in the right side of the above equality has valuation strictly smaller than the other three. Hence, by (3.7) and (3.8), one has ss′ ≠ 0 and

w(ss′) = w((c_{ρ}z_{ρ} ⊗ y_{ρ})(d_{ι}z_{ι} ⊗ y_{ι})) = w(s) + w(s′).

Therefore w is a valuation on E. Since E = E′ ⊗_{L′} L, the restriction of w to E′ is also a valuation. The uniqueness of w follows from its existence by [W_{1}]. □

Let D be a division algebra with a valuation. The residue division algebra of D is denoted by \overline{D}.

We keep the notations above. Now, suppose C is a division subalgebra of A and denote by Γ_{E} and Γ_{L} the corresponding value groups of E and L respectively. Furthermore, assume that E has an armature A. The diagram

\xymatrix{1 \ar[r] & L^{×} \ar[r] \ar[d]^{v} & E^{×} \ar[r] \ar[d]^{w} & E^{×}/L^{×} \ar[r] \ar@{.>}[d] & 1 \\ 0 \ar[r] & Γ_{L} \ar[r] & Γ_{E} \ar[r] & Γ_{E}/Γ_{L} \ar[r] & 0}

induces a homomorphism

w′ : A ⊂ E^{×}/L^{×} ⟶ Γ_{E}/Γ_{L}.

Put A_{0} = ker w′ and let a ∈ A_{0}. The above diagram shows that there exists a representative x_{a} of a such that w(x_{a}) = 0. Define

¯ : A_{0} ⟶ \overline{E}^{×}/\overline{L}^{×} = C^{×}/F^{×}

by

a = x_{a}L^{×} ⟼ ā = x̄_{a}F^{×}

where x_{a} is such that w(x_{a}) = 0 and x̄_{a} is the residue of x_{a}. If y_{a} is another representative of a such that w(y_{a}) = 0, there is h ∈ L^{×} with w(h) = 0 such that y_{a} = x_{a}h. Hence ȳ_{a} = x̄_{a}h̄, that is ȳ_{a} = x̄_{a}F^{×}, so ¯ is well defined. We have:

Proposition 3.6. Assume that C is a division algebra and A is an armature of E as above. Then

(1) The map ¯ : A_{0} ⟶ C^{×}/F^{×} is an injective homomorphism.

(2) The image \overline{A_{0}} of A_{0} is an armature of C over F.

Proof. (1) Let a = x_{a}L^{×} and b = x_{b}L^{×} be such that w(x_{a}) = w(x_{b}) = 0. Choosing x_{ab} = x_{a}x_{b}, we have w(x_{ab}) = 0. Therefore x̄_{ab} = x̄_{a}x̄_{b}; this shows that ¯ is a homomorphism.

Let c ∈ ker¯ with c ≠ 1 and let x_{c} ∈ E^{×} be a representative of c such that w(x_{c}) = 0. Let x̄_{c} = α ∈ F^{×}; then x_{c} = α + x′_{c} for some x′_{c} ∈ E with w(x′_{c}) > 0. The pairing ⟨ , ⟩ being nondegenerate on A, there exists d ∈ A such that ⟨d, c⟩ = ζ for some 1 ≠ ζ ∈ μ(F). Thus,

\overline{x_{d}x_{c}x_{d}^{−1}} = ζx̄_{c} = ζα.

On the other hand,

x_{d}x_{c}x_{d}^{−1} = x_{d}αx_{d}^{−1} + x_{d}x′_{c}x_{d}^{−1} = α + x_{d}x′_{c}x_{d}^{−1}.

Hence, we get

\overline{x_{d}x_{c}x_{d}^{−1}} = α since w(x_{d}x′_{c}x_{d}^{−1}) > 0; contradiction.

Therefore the map ¯ is injective. Consequently, we have |\overline{A_{0}}| = |A_{0}|.

(2) We first show that |A_{0}| ≤ dim_{F} C: since C = \overline{E}, it suffices to prove that (x̄_{a})_{a∈A_{0}} are linearly independent over F. Let ∑_{a∈A_{0}} λ_{a}x̄_{a} = 0, with λ_{a} ∈ F, be a zero linear combination such that the set

S = {a ∈ A_{0} | λ_{a} ≠ 0}

is not empty and of least cardinality. For s ∈ S, let x_{s} be a representative of s in E^{×} such that w(x_{s}) = 0. We have

x̄_{s}( ∑_{a∈A_{0}} λ_{a}x̄_{a})x̄_{s}^{−1} = ∑_{a∈A_{0}}⟨a, s⟩λ_{a}x̄_{a} = 0 = ∑_{a∈A_{0}} λ_{a}x̄_{a}.

Then the linear combination ∑_{a∈A_{0}}(1 − ⟨a, s⟩)λ_{a}x̄_{a} is zero and the number of nonzero coefficients is less than the cardinality of S because ⟨s, s⟩ = 1. Therefore, ⟨a, s⟩ = 1 for all a, s ∈ S; this implies that x̄_{s} and x̄_{s′} commute for all s, s′ ∈ S. It follows from [T_{1}, Lemma 1.5] that the elements x̄_{s}, for s ∈ S, are linearly independent; contradicting the fact that S is not empty. Combining with the part (1), we get

|\overline{A_{0}}| = |A_{0}| ≤ dim C = \frac{1}{n_{1} … n_{r}} |A|.

On the other hand, since |w′(A)| = \frac{|A|}{|A_{0}|}, we have |w′(A)| ≥ n_{1} … n_{r}. We already know that |w′(A)| ≤ n_{1} … n_{r} because w′(A) ⊂ Γ_{E}/Γ_{L} and |Γ_{E}/Γ_{L}| = n_{1} … n_{r}. It follows that |w′(A)| = n_{1} … n_{r} and |\overline{A_{0}}| = dim_{F} C. Since we showed that (x̄_{a})_{a∈A_{0}} are linearly independent, the subgroup \overline{A_{0}} is an armature of C over F. □

3.2. Proof of the main result. Recall that the division algebras Brauer equivalent to R′ and R are respectively E′ and E by Lemma 3.4.

Proof of Theorem 3.1. (i) ⇒ (ii): we give the proof for E; the proof for E′ follows because if E′ has an armature then E = E′ ⊗_{L′} L has an isomorphic armature. Assume that E has an armature A. Let c ∈ A and let c_{ρ}z_{ρ} ⊗ y_{ρ} be the leading term of a representative x_{c} of c in E. That is, x_{c} = c_{ρ}z_{ρ}⊗y_{ρ}+x′_{c} with w(x_{c}) = w(c_{ρ}z_{ρ}⊗y_{ρ}) and w(x′_{c}) > w(x_{c}). Define the map

ν : A ⟶ A^{×}/F^{×}

by

c = x_{c}.L^{×} ⟼ c_{ρ}z_{ρ}.F^{×}.

If y_{c} is another representative of c, we have y_{c} = ℓx_{c} for some ℓ ∈ L^{×}. Note that the leading term of y_{c} is the leading term of ℓ multiplied by c_{ρ}z_{ρ} ⊗ y_{ρ} (see the proof of Lemma 3.5); moreover, the leading term of ℓ lies in F^{×}. One deduces that ν(x_{c}.L^{×}) = ν(y_{c}.L^{×}). So ν is well-defined.

We show that ν(A) is an armature of A: first, we claim that ν is an injective homomorphism. Indeed, let a, b ∈ A with respective representatives x_{a} and x_{b}. Let c_{σ}z_{σ} ⊗ y_{σ} and d_{τ} z_{τ} ⊗ y_{τ} be the leading terms of x_{a} and x_{b} respectively. As showed in the proof of Lemma 3.5, the leading term of x_{a}x_{b} is (c_{σ}z_{σ} ⊗ y_{σ})(d_{τ} z_{τ} ⊗ y_{τ}). On the other hand, it follows by (3.1), (3.2), (3.3) and Lemma 3.4 that

(c_{σ}z_{σ} ⊗ y_{σ})(d_{τ} z_{τ} ⊗ y_{τ}) = c_{σ}(z_{σ}d_{τ} z_{σ}^{−1})f(σ, τ)c(σ, τ)z_{στ} ⊗ y_{στ}

for some f(σ, τ) ∈ L^{×} and some c(σ, τ) ∈ C^{×}, and z_{σ}d_{τ} z_{σ}^{−1} ∈ C.

Since x_{ab} = x_{a}x_{b} mod L^{×} (because a, b ∈ A), we may take x_{a}x_{b} as a representative of ab, so

x_{ab} = x_{a}x_{b} = c_{σ}(z_{σ}d_{τ} z_{σ}^{−1})c(σ, τ)z_{στ} ⊗ y_{στ} + x′_{ab} with w(x′_{ab}) > w(x_{ab}).

Hence, it follows by the definition of ν that

ν(ab) = c_{σ}(z_{σ}d_{τ} z_{σ}^{−1})c(σ, τ)z_{στ} mod F^{×}
= (c_{σ}z_{σ})(d_{τ} z_{τ}) mod F^{×}
= ν(a)ν(b).

Therefore, ν is a homomorphism.

For the injectivity, we start out by proving that the pairing ⟨ , ⟩ is an isometry for ν: by definition

⟨a, b⟩ = x_{a}x_{b}x_{a}^{−1}x_{b}^{−1} = ζ for some ζ ∈ μ(F).

Hence

x_{a}x_{b} = (c_{σ}z_{σ} ⊗y_{σ})(d_{τ} z_{τ} ⊗y_{τ})+(x_{a}x_{b})′ = ζx_{b}x_{a} = ζ(d_{τ} z_{τ} ⊗y_{τ})(c_{σ}z_{σ} ⊗y_{σ})+ζ(x_{b}x_{a})′,

with w((x_{a}x_{b})′) = w((x_{b}x_{a})′) > w(x_{a}x_{b}). Since y_{σ} and y_{τ} commute, it follows that

(c_{σ}z_{σ})(d_{τ} z_{τ}) = ζ(d_{τ} z_{τ})(c_{σ}z_{σ}).

Therefore

⟨ν(a), ν(b)⟩ = (c_{σ}z_{σ})(d_{τ} z_{τ})(c_{σ}z_{σ})^{−1}(d_{τ} z_{τ})^{−1} = ζ.

The pairing ⟨ , ⟩ is then an isometry for ν. Now, to see that ν is an injection, let a ∈ A be such that ν(a) = 1. Since ⟨ , ⟩ is an isometry for ν, for all b ∈ A, one has

1 = ⟨ν(a), ν(b)⟩ = ⟨a, b⟩.

We infer that a = 1 because the pairing is nondegenerate on A. It follows that ν(A) is an abelian subgroup of A^{×}/F^{×} isomorphic to A. Consequently, ν(A) is an armature of A^{×}/F^{×} isomorphic to M (≃ A by hypothesis) since deg E = deg A by Lemma 3.4.

Now, we prove that Kum(M/F) ⊂ ν(A). Since \overline{A_{0}} is an armature of C by Proposition 3.6, it follows by [TW, Lemma 2.5] that rad(\overline{A_{0}}) is an armature of the center of C which is M. The extension M/F being a Kummer extension, Examples 2.2 (b) indicates that rad(\overline{A_{0}}) = Kum(M/F). Therefore, we have Kum(M/F) ⊂ ν(A) since ν is the identity on A_{0}.

(ii) ⇒ (i): let B be an armature of A isomorphic to M and containing Kum(M/F) as a totally isotropic subgroup. We construct an isomorphic armature in E′. Note that if E′ has an armature, then E also has an armature. For each σ ∈ G, set

B_{σ} = {a ∈ B | ⟨a, b⟩ = σ(x_{b})x_{b}^{−1} for all b ∈ Kum(M/F)}.

One easily checks that B_{id} = Kum(M/F)^{⊥} and for a, c ∈ B_{σ}, ac^{−1} ∈ B_{id}. The sets B_{σ} are the cosets of B_{id} in B. So, we have the disjoint union B = ⨆_{σ∈G} B_{σ}. On the other hand, for a ∈ B_{σ} and b ∈ Kum(M/F), comparing the equality ⟨a, b⟩ = σ(x_{b})x_{b}^{−1} and the definition ⟨a, b⟩ = x_{a}x_{b}x_{a}^{−1}x_{b}^{−1} we get x_{a}x_{b} = σ(x_{b})x_{a}^{−1}; this implies B_{σ} ⊂ C^{×}z_{σ}/F^{×}. Now, let us denote

B′_{σ} = {(x_{a_{σ}} ⊗ y_{σ}).L^{′×} | a_{σ} ∈ B_{σ}} and B′ = ⨆_{σ∈G} B′_{σ}.

Note that B′_{σ} ⊂ C_{L′}^{×}(z_{σ} ⊗ y_{σ})/L^{′×} and we readily check that B′ is a subgroup of E^{′×}/L^{′×}. We claim that B′ is an armature of E′: since deg A = deg E′, it follows by the definition of B′ that |B′| = dim E′. Moreover, as in the part (2) of the proof of Proposition 3.6, one verifies that the representatives of the elements of B′ in E′ are linearly independent over L′. It remains to show that B′ is commutative. Let a_{σ} ∈ B_{σ} and d_{τ} ∈ B_{τ} with σ, τ ∈ G. By (3.3), y_{σ}y_{τ} = y_{τ} y_{σ} because f(σ, τ) = f(τ, σ). Furthermore, taking x_{a_{σ}}x_{d_{τ}} as a representative of a_{σ}d_{τ} (since B is an armature), we have x_{a_{σ}}x_{d_{τ}} = x_{a_{σ}d_{τ}} = x_{d_{τ}} x_{a_{σ}}. The commutativity of B′ follows; and therefore B′ is an armature of E′.

Using the same arguments as above, we see that the map B′ → B that carries (x_{a_{σ}} ⊗ y_{σ}).L^{′×} to x_{a_{σ}}.F^{×} is an isomorphism. Consequently, the armature B′ is also isomorphic to M. This concludes the proof. □

Proof of Corollary 3.2. (i) ⇒ (ii): as in the proof of Theorem 3.1, it is enough to give the proof for E. Assume that E decomposes into a tensor product of symbol algebras of degree p. Recall that if E decomposes into a tensor product of symbol algebras of degree p then E has an armature of exponent p (see [TW, Prop. 2.7]). It follows by Theorem 3.1 that A decomposes into a tensor product of symbol algebras of degree p. More precisely, if A is an armature of E of exponent p, we showed that ν(A) is an armature of A isomorphic to A and Kum(M/F) ⊂ ν(A). Now, let x_{1}, …, x_{r} ∈ A be such that k_{i} = F (x_{i}). The subgroup generated by (x_{i}F^{×}) for i = 1, …, r is Kum(M/F). The exponent of ν(A) being p, we may view ν(A) as a vector space over the field with p elements. Since M is a field, the elements e_{1} := x_{1}F^{×}, …, e_{r} := x_{r}F^{×} are linearly independent in ν(A). On the other hand, M being commutative, the subspace spanned by e_{1}, …, e_{r} is totally isotropic with respect to ⟨ , ⟩. It follows then by Proposition 2.3 that there are f_{1}, …, f_{r}, e_{r+1}, f_{r+1}, …, e_{n}, f_{n} in ν(A) such that {e_{1}, f_{1}, …, e_{n}, f_{n}} is a symplectic base of ν(A). Expressing f_{i} = y_{i}F^{×} for i = 1, …, r and ν(A) = A_{1} × … × A_{n}, where A_{i} = (e_{i}) × (f_{i}) for i = 1, …, n, we get

A ≃ (k_{1}, σ_{1}, δ_{1}) ⊗_{F} ⋯ ⊗_{F} (k_{r}, σ_{r}, δ_{r}) ⊗_{F} F [A_{r+1}] ⊗_{F} ⋯ ⊗_{F} F [A_{n}]

with δ_{i} = y_{i}^{p} for i = 1, …, r.

(ii) ⇒ (i): assume that A decomposes as

A ≃ (k_{1}, σ_{1}, δ_{1}) ⊗_{F} ⋯ ⊗_{F} (k_{r}, σ_{r}, δ_{r}) ⊗_{F} A_{r+1} ⊗_{F} ⋯ ⊗_{F} A_{n}

for some δ_{1}, …, δ_{r} ∈ F^{×} and some symbols subalgebras A_{r+1}, …, A_{n} of A. We give the proof for R. The same argument is valid for R′. We have

R = A ⊗_{F} (k_{1} ⊗_{F} L, σ_{1} ⊗ id, t_{1}) ⊗_{L} ⋯ ⊗_{L} (k_{r} ⊗_{F} L, σ_{r} ⊗ id, t_{r}) ∼ (k_{1} ⊗_{F} L, σ_{1} ⊗ id, δ_{1}t_{1}) ⊗_{L} ⋯ ⊗_{L} (k_{r} ⊗_{F} L, σ_{r} ⊗ id, δ_{r}t_{r}) ⊗_{F} A_{r+1} ⊗_{F} ⋯ ⊗_{F} A_{n}

(see for instance [D, §10]). Since this latter algebra has the same degree as A and deg(A) = deg(E) by Lemma 3.4, it is isomorphic to E. The proof is complete. □

4. Square-central elements

Let A be a central simple F -algebra of exponent 2 and let g ∈ A^{×} − F be a square-central element. The purpose of this section is to investigate conditions for g to be in a quaternion subalgebra of A and to give examples of tensor products of quaternion algebras containing a square-central element which is in no quaternion subalgebra.

4.1. The algebra A is not a division algebra. Here we distinguish two cases, according to whether g^{2} ∈ F^{×2} or g^{2} ∉ F^{×2}. Actually, we will not need to mention in the following proposition that A is not a division algebra because this is encoded by the fact that g ∈ A^{×} − F^{×} and g^{2} ∈ F^{×2}. Indeed, if g^{2} = λ^{2} with λ ∈ F^{×} then (g − λ)(g + λ) = 0; this means that A is not division.

Proposition 4.1. Let A be a central simple F -algebra and let g ∈ A^{×} − F^{×} be such that g^{2} = λ^{2}, λ ∈ F^{×}. The element g is in a quaternion subalgebra of A if and only if dim(g − λ)A = dim(g + λ)A. If the characteristic of F is 0, this condition holds if and only if the reduced trace Trd_{A}(g) of g is zero.

Proof. We can write A ≃ End_{D}(V) where D is a division algebra Brauer equivalent to A and V is some right D-vector space. Suppose there is a quaternion F -subalgebra Q of A such that g ∈ Q. Then A = Q ⊗ C_{A}Q, where C_{A}Q is the centralizer of Q in A. Since g^{2} = λ^{2} with λ ∈ F^{×}, we may identify Q with M_{2}(F) in such a way that g is the diagonal matrix diag(λ, −λ). Computations show that dim(g − λ)A = dim(g + λ)A = 2 dim C_{A}Q.

Conversely, suppose dim(g − λ)A = dim(g + λ)A. Let V_{+} and V_{−} be the λeigenspace and −λ-eigenspace of g respectively. For all u ∈ V, we have u = \frac{1}{2}(u + λ^{−1}g(u)) + \frac{1}{2}(u − λ^{−1}g(u)) ∈ V_{+} + V_{−} and V_{+} ∩ V_{−} = \{0\}. Hence V = V_{+} ⊕ V_{−}. Denote by r and s the dimensions of V_{+} and V_{−} respectively. Since g^{2} = λ^{2}, g is represented in A ≃ End_{D}(V) by the diagonal matrix diag(λ, …, λ, −λ, …, −λ) where the number of λ is r and the number of −λ is s. Computations show that dim(g − λ)A = s dim D and dim(g + λ)A = r dim D; therefore the hypothesis yields

r = s. The endomorphism f whose matrix is the block matrix f = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) , where

each block is an r × r matrix, anticommutes with g and is square-central. It follows that g lies in the split quaternion subalgebra of A generated by g and f.

Now suppose the characteristic of F is 0. The element g is represented by diag(λ, …, λ, −λ, …, −λ). So Trd_{A}(g) = (r −s)λ deg D and dim(g −λ)A = s dim D and dim(g + λ)A = r dim D. Therefore Trd_{A}(g) = 0 if and only if dim(g − λ)A = dim(g + λ)A. The proof is complete. □

If the characteristic of F is positive, the hypothesis on the trace does not suffice as we observe in the following counterexample:

Contrexample 4.2. Assume that F = 𝔽_{3}, the field with three elements, and take A = M_{8}(F). The diagonal matrix g = diag(1, …, 1, −1) is such that g^{2} = 1 and the trace of g is 0. But Proposition 4.1 shows that g is not in a quaternion subalgebra of A since dim(g + 1)A = 56 ≠ dim(g − 1)A = 8.

Proposition 4.3. Let A be a central simple F -algebra and let g ∈ A^{×} − F^{×} be such that g^{2} = a ∈ F^{×} − F^{×2}. The element g lies in a split quaternion subalgebra of A if and only if \frac{deg A}{ind(A)} is even.

Proof. If g ∈ M_{2}(F) ⊂ A, then A = M_{2}(F) ⊗ C where C is the centralizer of M_{2}(F) in A. Hence, \frac{deg A}{ind(A)} is even. Conversely, assume \frac{deg A}{ind(A)} is even. So, we may write

A ≃ M_{2}(F) ⊗ A′ for some algebra A′ Brauer equivalent to A. Set g′ = (\begin{matrix}0 & 1 \\ a & 0 \end{matrix});

we have g′ ∈ M_{2}(F) ⊂ A and g^{′2} = a. By the Skolem-Noether Theorem, g and g′ are conjugated. It follows that g is in a split quaternion subalgebra of A since g′ ∈ M_{2}(F). □

4.2. The algebra A is a division algebra. Let A be a division algebra and let x ∈ A^{×} − F^{×}. Recall that, A being a division algebra, we have necessarily x^{2} ∈ F^{×} − F^{×2}. Here we argue on the degree of the division algebra.

Degree 4. The following result is due to Albert and many proofs exist in the literature (see for instance [Ra], [LLT, Prop. 5.2], [Be, Thm. 4.1]). We propose the following proof for the reader’s convenience.

Proposition 4.4. Suppose A is a central simple algebra over F of degree 4 and exponent 2. Let x ∈ A^{×} − F^{×} be a square-central element with x^{2} ∉ F^{×2}. Then, x is in a quaternion F -subalgebra of A.

Proof. Note that F (x) is isomorphic to a quadratic extension of F since x^{2} ∈ F^{×} − F^{×2}. If A is not a division algebra, the result follows by Proposition 4.3. We assume that A is a division algebra. The centralizer C_{A}(x) of x in A is a quaternion algebra over F (x). The algebra C_{A}(x) is Brauer equivalent to A_{F (x)} (see for instance [P, §13.3]). Since

cor_{F (x)/F} [C_{A}(x)] = cor_{F (x)/F} (res_{F (x)/F} [A]) = 2[A] = 0

in Br(F) (see for instance [KMRT, (3.13)]), it follows from a result of Albert (see [KMRT, (2.22)]) that there is a quaternion algebra Q over F such that C_{A}(x) = Q ⊗_{F} F (x). Then, A = Q ⊗ C_{A}(Q) and the centralizer C_{A}(Q) of Q is a quaternion F -subalgebra of A containing x. □

Degree 8. Here we give an example of a tensor product of three quaternion algebras containing a square-central element which is in no quaternion subalgebra. This example is a private communication from Merkurjev to Tignol based on the following result:

Lemma 4.5 (Tignol). Let A be a division algebra over F of degree 8 and exponent 2. Let x ∈ A^{×} − F^{×} be such that x^{2} = a ∈ F^{×}. Then, there exists quaternion algebras Q_{1}, Q_{2}, Q_{3} such that M_{2}(A) ≃ Q_{1} ⊗ Q_{2} ⊗ Q_{3} ⊗ (a, y) for some y ∈ F.

Proof. It is shown in [J_{2}, Thm. 5.6.38] that M_{2}(A) is a tensor product of four quaternion algebras. The proof shows that one of these quaternion algebras can be chosen to contain x. □

Corollary 4.6 (Merkurjev). There exists a decomposable F -algebra of degree 8 and exponent 2 containing a square-central element which is in no quaternion subalgebra.

Proof. Let A be an indecomposable F -algebra of degree 8 and exponent 2 and let x ∈ A be such that x^{2} = a ∈ F^{×} with x ∉ F. Such an algebra A exists by [ART] and the existence of such an element x follows from a result of Rowen [J_{2}, Thm. 5.6.10]. Lemma 4.5 indicates that M_{2}(A) ≃ Q_{1} ⊗ Q_{2} ⊗ Q_{3} ⊗ (a, y) for some y ∈ F. Set D = Q_{1} ⊗ Q_{2} ⊗ Q_{3}. We claim that D is a division algebra. Indeed, if D is not a division algebra then D ≃ M_{2}(D′) where D′ is an algebra of degree 4 and exponent 2. Since an exponent 2 and degree 4 central simple algebra is always decomposable by a well-known result of Albert (see for instance [Ra]), we deduce that A is isomorphic to a product of quaternion algebras; this contradicts our hypothesis. Hence D is a division algebra. Since the algebras D_{F (\sqrt{a})} and A_{F (\sqrt{a})} are isomorphic and A_{F (\sqrt{a})} is not a division algebra, D_{F (\sqrt{a})} is not a division algebra. Then, by [A, Thm. 4.22] the algebra D contains an element α such that α^{2} = a with α ∉ F. Assume that D contains a quaternion subalgebra containing α, say (a, b) for some b ∈ F. The centralizer of (a, b) in D is an algebra of exponent 2 and degree 4. Thus, we have D ≃ H_{1} ⊗ H_{2} ⊗ (a, b) where H_{i} are quaternion algebras. It follows that M_{2}(A) ≃ H_{1} ⊗ H_{2} ⊗ (a, b) ⊗ (a, y) ≃ M_{2}(H_{1} ⊗ H_{2} ⊗ (a, yb)). Whence A ≃ H_{1} ⊗ H_{2} ⊗ (a, yb); contradiction. The algebra D satisfies the required conditions. □

Degree 2^{n}, n > 3. In this part, we generalize Corollary 4.6: we are going to construct a tensor product of n (with n > 3) quaternion algebras containing a square central element which is not in a quaternion subalgebra. To do this, we use valuation theory.

Let L = F ((t_{1}))((t_{2})) be the iterated Laurent power series field where t_{1}, t_{2} are independent indeterminates over F and let D be a division F -algebra. Set D′ = D ⊗ (t_{1}, t_{2})_{L} and let i, j ∈ D′ be such that i^{2} = t_{1}, j^{2} = t_{2} and ij = −ji. Since i^{2} = t_{1} and j^{2} = t_{2}, every element f ∈ D′ can be written as an iterated Laurent series in i and j with coefficients in D:

f = ∑_{β≥n} ∑_{α≥m_{β}} d_{α,β}i^{α}j^{β} with d_{α,β} ∈ D and n, m_{β} ∈ ℤ.

Define v : D^{′×} ⟶ (\frac{1}{2}ℤ)^{2} (where (\frac{1}{2}ℤ)^{2} is ordered lexicographically from right-to-left) by

v(f) = inf \{(\frac{α}{2}, \frac{β}{2} ) | d_{α,β} ≠ 0\}.

Computations show that v is a valuation on D′. Actually, v is the unique extension of the (t_{1}, t_{2})-adic valuation on L (which is Henselian). As in the previous section, for f ∈ D^{′×}, the leading term of f is defined to be

ℓ(f) = d_{m,n}i^{m}j^{n} where (m, n) = v(f).

Straightforward computations show that

(i) ℓ(fg) = ℓ(f)ℓ(g), for f, g ∈ D^{′×}; (ii) ℓ(d) = d, for d ∈ D^{×}; (iii) ℓ(z) ∈ L, for z ∈ L^{×}.

We have the following generalization of Corollary 4.6:

Proposition 4.7. Let D be a division algebra over F. Let x ∈ D^{×} − F be a squarecentral element which is in no quaternion subalgebra of D. Then D ⊗ (t_{1}, t_{2})_{L} has no quaternion subalgebra containing x.

Proof. Suppose there is y ∈ D ⊗ (t_{1}, t_{2})_{L} such that y^{2} ∈ L^{×} and xy = −yx. Let ℓ(y) = di^{α}j^{β} with d ∈ D^{×} and α, β ∈ ℤ. We have ℓ(y)i^{−α}j^{−β} = d and d^{2} ∈ F^{×}. Since xy = −yx we have ℓ(x)ℓ(y) = −ℓ(y)ℓ(x), that is, xℓ(y) = −ℓ(y)x. Hence, d anticommutes with x; contradiction with the choice of x. □

4.3. An application. Corollary 4.6 implies that if F is the center of an indecomposable algebra of degree 8 and exponent 2, then there exist a decomposable division algebra of degree 8 and exponent 2 containing a square-central element which is not in a quaternion subalgebra. Conversely, let D be a division algebra of degree 8 and exponent 2 over F and let F (\sqrt{a}) ⊂ D be a quadratic field extension of F such that F (\sqrt{a}) is not in a quaternion subalgebra of D (the algebra D could be decomposable). Theorem 3.2 shows that the division algebra Brauer equivalent to D ⊗_{F} (a, t)_{F (t)} is an indecomposable algebra of degree 8 and exponent 2 over F (t).

As an application, we are going now to give an example of indecomposable algebra of degree 8 and exponent 2 over a field of 2-cohomological dimension 4. Let F be a field of characteristic different from 2 and let us denote K = F (\sqrt{a}). Let B be a biquaternion algebra over K with trivial corestriction, cor_{K/F} (B) = 0. In [Ba], it is associated with B a degree three cohomological invariant δ_{K/F} (B) with value in H^{3}(F, μ_{2})/ cor_{K/F} ((K^{×}) · [B]) where H^{3}(F, μ_{2}) is the third Galois cohomology group of F with coefficients in μ_{2} = {±1}. It is also shown in [Ba] that B has a descent to F (that is, B = B_{0} ⊗_{F} K for some biquaternion algebra B_{0} defined over F) if and only if δ_{K/F} (B) = 0.

Now, let D be a central simple algebra of degree 8 and exponent 2 over F containing K such that K is not in a quaternion subalgebra of D. That also means δ_{K/F} (C_{D}K) ≠ 0 where C_{D}K denotes the centralizer of K in D. Consider the following Merkurjev extension 𝕄 of F:

F = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{∞} = ⋃_{i} F_{i} =: 𝕄

where the field F_{2i+1} is the maximal odd degree extension of F_{2i}; the field F_{2i+2} is the composite of all the function fields F_{2i+1}(π), where π ranges over all 4-fold Pfister forms over F_{2i+1}. The arguments used by Merkurjev in [M] show that the 2-cohomological dimension cd_{2}(𝕄) ≤ 3. We have the following result:

Theorem 4.8. The algebra D and the field 𝕄 are as above. The division algebra Brauer equivalent to

D_{𝕄} ⊗_{𝕄} (a, t)_{𝕄(t)}

is indecomposable of degree 8 and exponent 2 over 𝕄(t), where t is an indeterminate.

Proof. Put B = C_{D}K. As observed in the proof of [Ba, Thm. 1.3] the 2-cohomological dimension of 𝕄 is exactly 3. It follows from [Ba, Prop. 4.7] and a result of Merkurjev (see Theorem A.9 of [Ba]) that the scalar extension map

\frac{H^{3}(F, μ_{2})}{cor_{K/F} ((K^{×}) · [B])} ⟶ \frac{H^{3}(𝕄, μ_{2})}{cor_{𝕄(\sqrt{a})/𝕄}(𝕄(\sqrt{a})^{×} · [B_{𝕄(\sqrt{a})}])}

is an injection. So, δ_{𝕄(\sqrt{a})/𝕄}(B) ≠ 0 since δ_{K/F} (B) ≠ 0. Hence the extension 𝕄(\sqrt{a}) is not in a quaternion subalgebra of D_{𝕄}. Therefore the division algebra Brauer equivalent to D_{𝕄} ⊗_{𝕄} (a, t)_{𝕄(t)} is an indecomposable algebra of degree 8 and exponent 2 over 𝕄(t) by Corollary 3.2; as desired. □

Acknowledgements. This work is a generalization of some results of my PhD thesis. I gracefully thank my advisors A. Quéguiner-Mathieu and J.-P. Tignol for their support, ideas and for sharing their knowledge. I thank A. S. Merkurjev for providing Corollary 4.6 (a private communication to J.-P. Tignol).

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ICTEAM Institute, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium

E-mail address: barry.demba@gmail.com