  Given a field $F$, an \'etale extension $L/F$ and an Azumaya algebra $A/L$,
one knows that there are extensions $E/F$ such that $A \otimes_F E$ is a split
algebra over $L \otimes_F E$. 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 $L/F$ 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
also show that for any odd prime number $p$, the generic algebra of index
$p^n$, and exponent $p$ cannot be expressed nontrivially as the corestriction
of an algebra over any extension field if $n < p^2$.
