SPLITTING FIELDS OF CENTRAL SIMPLE ALGEBRAS OF EXPONENT TWO

KARIM JOHANNES BECHER

Abstract. By Merkurjev’s Theorem every central simple algebra of exponent two is Brauer equivalent to a tensor product of quaternion algebras. In particular, if every quaternion algebra over a given field is split, then there exists no central simple algebra of exponent two over this field. This note provides an independent elementary proof for the latter fact. Classification (MSC 2010): 11E04, 11E81, 16H05, 16K20, 16K50 Keywords: Brauer group, field, central simple algebra, quaternion algebra, splitting field, index, exponent, 2-extension, involution

Central simple algebras and the Brauer group of a field are a classical topic in algebra. The aim of this note is to give an elementary proof for the existence of certain splitting fields for central simple algebras of exponent two.

Let K be a field. Let A be a central simple K-algebra. We say that A is split if it is a matrix algebra over K. The index of A is defined as the degree of the unique K-division algebra Brauer equivalent to A. The exponent of A is the order of the class of A in the Brauer group of A.

By [1, Theorem 4.5.13] the exponent divides the index and both numbers have the same prime factors. Examples can easily be constructed showing that any pair of natural numbers fulfilling these two constraints occur as the index and the exponent of a central simple algebra over some field.

By [1, Corollary 4.5.9] the index of A is equal to the smallest degree [L : K] of a finite separable field extension L/K such that the L-algebra A_{L} = A ⊗_{K} L is split. Apart from this condition on the degree, little is known about the finite extensions that split A. For example, if A has prime index p, it is an open problem (for p > 3) whether there exists a cyclic extension of degree p of K that splits A. The only way to obtain general statements on the existence of splitting fields of a certain type based on the index and the exponent seems to be the Merkurjev-Suslin Theorem [1, Theorem 2.5.7], a generalisation of Merkurjev’s result from [3] on central simple algebras of exponent two.

Let Br_{2}(K) denote the 2-torsion part of Br(K). Merkurjev’s Theorem [1, Theorem 1.5.8] tells us that Br_{2}(K) is generated by the classes of the K-quat-ernion algebras. (By [1, Theorem 9.1.4] this statement also holds in characteristic 2, where it is due to Teichmüller.) There are different proofs of Merkurjev’s Theorem in the literature (see the references in the introduction of [1, Chap. 8]), but all of them are quite involved.

Date: 18 March, 2016.

This work was supported by the Deutsche Forschungsgemeinschaft (project The Pfister Factor Conjecture in Characteristic Two, BE 2614/4) and the FWO Odysseus programme (project Explicit Methods in Quadratic Form Theory).

Merkurjev’s Theorem implies that any element of Br_{2}(K) is split by a finite Galois extension with 2-elementary abelian Galois group, so in particular by a 2-extension. In some applications to the study of algebras with involution and their cohomological invariants, only the latter fact is needed. This motivates looking for a more elementary argument for this consequence. This note provides such an argument, fitting into the basic part of the theory of central simple algebras and their involutions.

A finite extension L/K is called a 2-extension if it is separable and if the degree of its normal closure is a power of 2. By classical Galois theory and the fact that 2-groups are solvable, this is equivalent to the existence of r ∈ ℕ and a tower of fields (L_{i})_{i=0}^{r} with L_{0} = K, L_{r} = L, and where L_{i}/L_{i−1} is a separable quadratic extension for i = 1, …, r.

Lemma 1. Let L/K be a finite 2-extension. If every K-quaternion algebra is split, then every L-quaternion algebra is split.

Proof: Suppose first that [L : K] = 2 and consider an L-quaternion algebra Q. Let B denote the corestriction (norm) of Q with respect to L/K as defined in [2, §3.B]. Then B is a central simple K-algebra of degree 4 and exponent at most 2. By a theorem due to Albert [2, §16], B is isomorphic a tensor product of two K-quaternion algebras. Assume now that every K-quaternion algebra is split. Then B is split. In view of [2, (2.22) and (3.1)] this implies that Q is extended from a K-quaternion algebra. By the assumption this K-quaternion algebra is split. Hence Q is split.

This argument shows the statement in the case of a separable quadratic extension L/K. The general case then follows by induction, since any finite 2-extension is reached by a finite tower of separable quadratic extensions. □

Let L/K be a field extension. We denote by Br(L/K) the kernel of the natural homomorphism Br(K) ⟶ Br(L). For an element α ∈ Br(K) we denote by α_{L} its image in Br(L), and we say that L/K splits α or that α splits over L if α_{L} = 0. The index of α ∈ Br(K), denoted ind(α), is the minimal degree of a field extension L/K such that α_{L} = 0.

Lemma 2. Let L/K be a finite 2-extension. If every K-quaternion algebra is split, then Br(L/K) = 0.

Proof: Let r ∈ ℕ and let (L_{i})_{i=0}^{r} be a tower of fields such that L_{0} = K, L_{r} = L and L_{i}/L_{i−1} is a separable quadratic extension for i = 1, …, r. Assume that there exists a nontrivial element α ∈ Br(L/K). Then for some s ∈ {1, …, r} we have that α_{L_{s−1}} ≠ 0 and α_{L_{s}} = 0. Since [L_{s} : L_{s−1}] = 2 it follows that ind(α_{L_{s−1}}) = 2. Set L′ = L_{s−1}. Then L′/K is a finite 2-extension and α_{L′} is the class of a non-split L′-quaternion algebra. By Lemma 1 it follows that there exists a non-split K-quaternion algebra. □

We denote by char(K) the characteristic of K. We further write K^{×} for the multiplicative group of K and K^{×2} for the subgroup of non-zero squares.

Proposition. Assume that K^{×} = K^{×2} and that K is not algebraically closed. Let L/K be a proper finite field extension of K of minimal degree. Then Br(M/L) = 0 for every finite 2-extension M/L.

Proof: Consider an arbitrary L-quaternion algebra Q. If char(K) = 2, then as K^{×} = K^{×2} we have that K is perfect, hence L is perfect, and as char(L) = 2 it follows that Q is split. Suppose now that char(K) ≠ 2.

The hypothesis that L/K has minimal degree among all proper field extensions of K implies that every polynomial f ∈ K[T] with deg(f) < [L : K] is split. We fix an element x ∈ L\K. Then L = K[x], whereby Q ≃ (f(x), g(x))_{L} for two nonzero polynomials f, g ∈ K[T] of degree strictly smaller than [L : K]. Hence f and g are split. Since K^{×} = K^{×2} we may choose f and g to be monic. If one of f or g is constant then it is equal to 1, whereby Q is split. As −1 ∈ K^{×2} we also have that Q ≃ (f(x), −g(x))_{L}. If f and g are linear, then f − g is a square in K, and it follows that Q is split. In the general case, writing f = ∏_{i=1}^{m} f_{i} and g = ∏_{i=1}^{n} g_{i} with m, n ∈ ℕ and monic linear polynomials f_{1}, …, f_{m}, g_{1}, …, g_{n} ∈ K[T], we conclude that (f_{i}(x), g_{j}(x))_{L} is split for any i ∈ {1, …, m} and any j ∈ {1, …, n}. By the bilinearity of the quaternion pairing L^{×} × L^{×} ⟶ Br(L), it follows that Q is split.

Hence, we have shown that every L-quaternion algebra is split. By Lemma 2 this yields the statement. □

We say that K is quadratically closed if every quadratic polynomial in K[T] is split, or equivalently, if K admits no quadratic field extension.

Lemma 3. Assume there exists an element α ∈ Br(K) satisfying the following:

(a) For any finite 2-extension L/K we have α_{L} ≠ 0.

(b) For any proper finite field extension K′/K, there exists a 2-extension L/K′ such that α_{L} = 0.

Then K is quadratically closed.

Proof: Suppose that K is not quadratically closed. We fix a quadratic field extension K′/K. By (b) there exists a finite 2-extension L′/K′ with α_{L′} = 0. Let K′′ denote the relative separable closure of K inside L′. Then K′′/K is a finite 2-extension and [L′ : K′′] ≤ [K′ : K] = 2. Since α_{L′} = 0 it follows that ind(α_{K′′}) ≤ 2. If ind(α_{K′′}) = 2, then we may choose a separable quadratic extension L/K′′ such that α_{L} = 0. If ind(α) = 1 then we set L = K′′. In either case L/K is a finite 2-extension such that α_{L} = 0, in contradiction to (a). □

Lemma 4. Let α ∈ Br_{2}(K) with α ≠ 0. There exists a finite field extension L/K such that [L : K] = \frac{1}{2}ind(α) and ind(α_{L}) = 2.

Proof: Let D be the central K-division algebra whose class in Br(K) is α. Since α + α = 0 ≠ α, it follows by [2, (2.8) and (3.1)] that there exists a symplectic involution σ on D. Let L/K be a separable field extension contained in D with σ|_{L} = id_{L} and maximal for these properties. Using [2, (4.12)] it follows that [L : K] = \frac{1}{2}ind(α) and thus ind(α_{L}) = 2. □

We are ready to prove the main statement.

Theorem. Every element of Br(K) of order a power of 2 is split by a finite 2-extension of K.

Proof: Consider an element α ∈ Br(K) of order 2^{r} where r ∈ ℕ. If 2α is split by a 2-extension K′/K, then α_{K′} ∈ Br_{2}(K′), and if furthermore α_{K′} is split by a 2-extension L/K′, then L/K is a 2-extension splitting α. Hence, the statement follows by induction on r from the case where α ∈ Br_{2}(K).

Suppose now that α ∈ Br_{2}(K) and that α is not split by any finite 2-extension of K. Using Zorn’s Lemma we obtain an algebraic field extension K′/K that is maximal for the property that α_{K′} is not split by any finite 2-extension of K′. By Lemma 3 this maximality implies that K′ is quadratically closed. On the other hand α_{K′} ≠ 0, whereby K′ is not algebraically closed. Let L/K′ be a proper finite field extension of minimal degree. Then 1 < [L : K′] < ind(α_{K′}), by Lemma 4. Hence, α_{L} ≠ 0 and by the maximality of K′ there exists a finite 2-extension M/L such that α_{M} = 0. This contradicts the Proposition. □

Corollary. Assume that every K-quaternion algebra is split. Then every element of Br(K) has odd order.

Proof: Since Br(K) is a torsion group it suffices to show that Br_{2}(K) = 0. Consider α ∈ Br_{2}(K). By the Theorem we have that α ∈ Br(L/K) for some finite 2-extension L/K. Since every K-quaternion algebra is split, we obtain by Lemma 2 that Br(L/K) = 0, whereby α = 0. □

It would be desirable to have a more explicit version of the above Theorem that provides a bound on the degree of a splitting 2-extension for α ∈ Br_{2}(K) in terms of ind(α). The proof presented here does not yield any such bound, as it uses Zorn’s Lemma.

Question 1. Is every element α ∈ Br_{2}(K) split by a 2-extension L/K of degree [L : K] = ind(α)?

Equivalently, one may ask whether any finite-dimensional division algebra of exponent two contains a separable quadratic extension of its centre.

The answer to Question 1 is known to be positive when ind(α) ≤ 8; for ind(α) = 8 this follows from a theorem due to Rowen (cf. [5, p. 279, Exercise 32] and [4, Theorem 1]).

We mention another open problem related to the above Corollary.

Question 2. If K is quadratically closed, does it follow that Br_{2}(L) = 0 for every finite field extension L/K?

References

[1] P. Gille and T. Szamuely. Central simple algebras and Galois cohomology. Cambridge University Press (2006).

[2] M.-A. Knus, A.S. Merkurjev, M. Rost, and J.-P. Tignol. The book of involutions. American Mathematical Society Colloquium Publications 44. American Mathematical Society, Providence, RI, 1998.

[3] A. S. Merkurjev. On the norm residue symbol of degree 2. (Russian) Dokl. Akad. Nauk. SSSR 261 (1981): 542–547. English translation: Soviet Math. Dokl. 24 (1981): 546–551.

[4] L.H. Rowen, Division algebras of exponent 2 and characteristic 2, J. Algebra 90 (1984) 71–83.

[5] L. H. Rowen. Ring Theory, Vol. II. Pure and Applied Mathematics, 128. Academic Press, Inc., Boston, MA, 1988.

Departement Wiskunde–Informatica, Universiteit Antwerpen, Belgium

   E-mail address: becher@maths.ucd.ie