Three theorems on common splitting fields of central simple algebras

Let A(1),...,A(n) be central simple algebras over a field F. Suppose that we possess information on the Schur indexes of some tensor products of (some tensor powers of) the algebras. What can be said tin general) about possible degrees of finite field extensions of F splitting the algebras? In Part I, we prove a positive result of that kind. In Part II, we prove a negative result. In Part III, we develop a general approach.