Submission: 2007. Apr 25
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 $n < p^2$.
2000 Mathematics Subject Classification: 16K20
Keywords and Phrases: central simple algebra, division algebra, splitting field, Azumaya algebra, corestriction, Severi-Brauer variety
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