Prescribed Behavior of Central Simple Algebras after Scalar Extension

1. Let A1, . . . ,An be central simple disjoint algebras over a field F . Let also li|exp(Ai), mi|ind(Ai), li|mi, and for each i = 1, . . . , n, let li and mi have the same sets of prime divisors. Then there exists a field extension E/F such that exp(AiE) = li and ind(AiE) = mi, i = 1, . . . , n. 2. Let A be a central simple algebra over a field K with an involution τ of the second kind. We prove that there exists a regular field extension E/K preserving indices of central simple K-algebras such that A⊗K E is cyclic and has an involution of the second kind extending τ . Introduction and motivations This paper is a continuation of [18] where some properties of central simple algebras after scalar extensions were examined. In [18] we solved two problems. 1. For a given central simple K-algebra A, some K-variety X was constructed such that for a field extension L/K the variety X has an L-rational point iff A ⊗K L has some prescribed properties (e.g., being a symbol-algebra). 2. For a given central simple K-algebra A, a regular field extension E/K was constructed preserving indices of all central simple K-algebras, such that A⊗KE becomes a cyclic algebra. Note that if a field extension E/K preserves indices of all central simple K-algebras then E/K preserves exponents for all such K-algebras, but in some applications one needs to reduce exponents and indices of algebras in a prescribed manner. Below we fix the following notations and conventions. Let A be a finite dimensional central simple algebra over a field F . By Wedderburn’s theorem, there is a unique integer m ≥ 1 and a central division F -algebra D which is unique up to F -isomorphism such that A ∼= Mm(D). The degree of A is defined by deg(A) = √ dimF A, the index of A is said to be ind(A) = deg(D). Two central simple F -algebras A = Mm(D) and A′ = Mm′(D) are said to be Brauer equivalent if D ∼= D′. In this case we write A ∼ A′ and denote the equivalence class of A by [A]. The tensor product of central simple algebras defines an abelian group structure on this set of equivalence classes, called the Brauer group of F and denoted by Br(F ). The inverse of the class [A] is induced by the opposed algebra Aop of A. Am will denote the central simple algebra A⊗ · · · ⊗ A (m times). Date: October 9, 2011. The second and the third authors are grateful to the Department of Mathematics at the University of Bielefeld for the hospitality during the preparation of the paper. 1 2 REHMANN, TIKHONOV, YANCHEVSKĬI The neutral element is defined by the class A ∼ F , in this case we write A ∼ 1. The exponent exp(A) of A in Br(F ) is the order of [A] in Br(F ). It is known that exp(A) and ind(A) have the same prime divisors and exp(A)|ind(A) [17, §14.4, Prop. b]. For a field extension K/F , AK will denote the K-algebra A⊗F K. If [K : F ] is coprime to ind(A), then ind(AK) = ind(A) [17, §13.4, Prop.]. Let us recall three special types of central simple algebras: Crossed products (L/F,Gal(L/F ), f). Let L/F be a Galois field extension, Gal(L/F ) its Galois group and f a 2-cocycle of Gal(L/F ) with values in L. Then the left L-module with L-base {uτ}τ∈Gal(L/F ) and multiplication table usl = l us for l ∈ L, usut = f(s, t)ust for any s, t ∈ Gal(L/F ) is a central simple F -algebra and denoted by (L/F,Gal(L/F ), f). Cyclic algebras (E/F, σ, a). They are a special form of crossed products. Let E/F be a cyclic field extension of degree n, σ a generator of Gal(E/F ) and a ∈ F . Then (E/F, σ, a) is a left E-module with E-base {uσ}i=1,...,n and multiplication table: uσc = c iuσ and uσ = a for any i = 0, . . . , n− 1 and c ∈ E. The corresponding cocycle is the following ca(σ , σ) = { 1, if i+ j < [E : F ]; a, if i+ j ≥ [E : F ]. Symbol algebras (a, b)n. These algebras also have a simple set of generators and defining relations. Let ρn ∈ F be a primitive root of unity of degree n and a, b ∈ F . Then (a, b)n is an n -dimensional vector F -space with an F -base {AB}i,j=1,...,n and multiplication table AB = ρ nB A, A = a, B = b. Following some arguments from [12] we prove in this paper, for disjoint algebras (see the Definition 1.1 below), the following Theorem 1. Let A1, . . . ,An be central simple disjoint algebras over F . Let also li|exp(Ai), mi|ind(Ai), li|mi such that, for each i = 1, . . . , n, both numbers li, mi have the same prime divisors. Then there exists a regular finitely generated field extension E/F such that exp(AiE) = li and ind(AiE) = mi, i = 1, . . . , n. The remaining part of the paper is devoted to algebras with involutions. Using ideas similar to those in [18] we prove the following Theorem 2. Let A be a central simple algebra over a field K with an involution τ of the second kind. Then there exists a regular field extension E/K preserving indices of central simple K-algebras such that AE is cyclic and has an involution of the second kind extending τ .