Corestrictions of algebras and splitting fields

Abstract

Given a field F F , an étale extension L / F L/F and an Azumaya algebra A / L A/L , one knows that there are extensions E / F E/F such that A ⊗ F E A \otimes _F E is a split algebra over L ⊗ F E L \otimes _F E . In this paper we bound the degree of a minimal splitting field of this type from above and show that our bound is sharp in certain situations, even in the case where L / F L/F is a split extension. This gives in particular a number of generalizations of the classical fact that when the tensor product of two quaternion algebras is not a division algebra, the two quaternion algebras must share a common quadratic splitting field. In another direction, our constructions combined with results of Karpenko (1995) also show that for any odd prime number p p , the generic algebra of index p n p^n and exponent p p cannot be expressed nontrivially as the corestriction of an algebra over any extension field if n > p 2 n > p^2 .