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Transactions of the American Mathematical Society

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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
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$.

References [Enhancements On Off] (What's this?)

  • [Art82] M. Artin.
    Brauer-Severi varieties.
    In Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), pages 194-210. Springer, Berlin, 1982. MR 657430 (83j:14015)
  • [Bla91] Altha Blanchet.
    Function fields of generalized Brauer-Severi varieties.
    Comm. Algebra, 19(1):97-118, 1991. MR 1092553 (92c:14052)
  • [Gro68] Alexander Grothendieck.
    Le groupe de Brauer. I. Algèbres d'Azumaya et interprétations diverses.
    In Dix Exposés sur la Cohomologie des Schémas, pages 46-66. North-Holland, Amsterdam, 1968. MR 0244269 (39:5586a)
  • [Jac96] Nathan Jacobson.
    Finite-dimensional division algebras over fields.
    Springer-Verlag, Berlin, 1996. MR 1439248 (98a:16024)
  • [Kar95] Nikita A. Karpenko.
    Torsion in $ {\rm CH}\sp 2$ of Severi-Brauer varieties and indecomposability of generic algebras.
    Manuscripta Math., 88(1):109-117, 1995. MR 1348794 (96g:14007)
  • [Kar99] Nikita A. Karpenko.
    Three theorems on common splitting fields of central simple algebras.
    Israel J. Math., 111:125-141, 1999. MR 1710735 (2000k:16020)
  • [Kra] Daniel Krashen.
    Zero cycles on homogeneous varieties.
    submitted for publication, arXiv:math.AG/0501399.
  • [MPW96] A. S. Merkurjev, I. A. Panin, and A. R. Wadsworth.
    Index reduction formulas for twisted flag varieties. I.
    $ K$-Theory, 10(6):517-596, 1996. MR 1415325 (98c:16018)
  • [Sch84] W. H. Schikhof.
    Ultrametric calculus, volume 4 of Cambridge Studies in Advanced Mathematics.
    Cambridge University Press, Cambridge, 1984.
    An introduction to $ p$-adic analysis. MR 791759 (86j:11104)
  • [Ser92] Jean-Pierre Serre.
    Topics in Galois theory.
    Jones and Bartlett Publishers, Boston, MA, 1992.
    Lecture notes prepared by Henri Damon [Henri Darmon], With a foreword by Darmon and the author. MR 1162313 (94d:12006)

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Additional Information

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

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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