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Corestrictions of algebras and splitting fields


Author: Daniel Krashen
Journal: Trans. Amer. Math. Soc. 362 (2010), 4781-4792
MSC (2010): Primary 16K20
DOI: https://doi.org/10.1090/S0002-9947-10-04967-6
Published electronically: April 26, 2010
MathSciNet review: 2645050
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Abstract: Given a field $ F$, an étale 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 (1995) 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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Additional Information

Daniel Krashen
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602

DOI: https://doi.org/10.1090/S0002-9947-10-04967-6
Received by editor(s): May 3, 2007
Received by editor(s) in revised form: November 18, 2008
Published electronically: April 26, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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