Corestrictions of algebras and splitting fields
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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
- MR Author ID: 728218
- ORCID: 0000-0001-6826-9901
- Received by editor(s): May 3, 2007
- Received by editor(s) in revised form: November 18, 2008
- Published electronically: April 26, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 4781-4792
- MSC (2010): Primary 16K20
- DOI: https://doi.org/10.1090/S0002-9947-10-04967-6
- MathSciNet review: 2645050