Prescribed behavior of central simple algebras after scalar extension

Abstract

1. Let A 1 , … , A n be central simple disjoint algebras over a field F. Let also l i | exp ( A i ) , m i | ind ( A i ) , l i | m i , and for each i = 1 , … , n , let l i and m i have the same sets of prime divisors. Then there exists a field extension E / F such that exp ( A i E ) = l i and ind ( A i E ) = m i , 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.