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⊗KE is cyclic and has an involution of the second kind extending τ.