# Corestrictions of algebras and splitting fields - Mathematics Rings and Algebras

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Abstract: 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 splitalgebra over $L \otimes F E$. In this paper we bound the degree of a minimalsplitting field of this type from above and show that our bound is sharp incertain situations, even in the case where $L-F$ is a split extension. Thisgives in particular a number of generalizations of the classical fact that whenthe tensor product of two quaternion algebras is not a division algebra, thetwo quaternion algebras must share a common quadratic splitting field.In another direction, our constructions combined with results of Karpenkoalso show that for any odd prime number $p$, the generic algebra of index$p^n$, and exponent $p$ cannot be expressed nontrivially as the corestrictionof an algebra over any extension field if $n < p^2$.

Autor: Daniel Krashen

Fuente: https://arxiv.org/