Corestrictions of algebras and splitting fields

Details Corestrictions of algebras and splitting fields Corestrictions of algebras and splitting fields

2007-04-25

arxiv.org/abs/0704.3443v1

Mathematics

Rings and Algebras

13 pages

Scientific paper

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 $n < p^2$.

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