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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Corestrictions of algebras and splitting fields
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by Daniel Krashen PDF
Trans. Amer. Math. Soc. 362 (2010), 4781-4792 Request permission

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