Prescribed behavior of central simple algebras after scalar extension

Abstract

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 .