Transfer of quadratic forms and of quaternion algebras over quadratic field extensions

Transfer of quadratic forms and of quaternion algebras over quadratic field extensions Arch. Math. 2018 Springer International Publishing AG, part of Springer Nature Archiv der Mathematik https://doi.org/10.1007/s00013-018-1198-5 Transfer of quadratic forms and of quaternion algebras over quadratic field extensions Karim Johannes Becher, Nicolas Grenier-Boley, and Jean-Pierre Tignol Abstract. Two different proofs are given showing that a quaternion al- gebra Q defined over a quadratic ´ etale extension K of a given field has a corestriction that is not a division algebra if and only if Q contains a quadratic algebra that is linearly disjoint from K. This is known in the case of a quadratic field extension in characteristic different from two. In the case where K is split, the statement recovers a well-known result on biquaternion algebras due to Albert and Draxl. Mathematics Subject Classification. 11E04, 11E81, 12G05, 16H05. Keywords. Isotropy, Witt index, Corestriction, Albert form, Character- istic two. 1. Introduction. A well-known theorem of Albert states that if a tensor prod- uct of two quaternion division algebras Q , Q over a field F of characteristic 1 2 different from 2 is not a division algebra, then there exists a quadratic ex- tension L of F that embeds as a subfield in Q and in Q ; see [6, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archiv der Mathematik Springer Journals

Transfer of quadratic forms and of quaternion algebras over quadratic field extensions

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Publisher
Springer International Publishing
Copyright
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Mathematics, general
ISSN
0003-889X
eISSN
1420-8938
D.O.I.
10.1007/s00013-018-1198-5
Publisher site
See Article on Publisher Site

Abstract

Arch. Math. 2018 Springer International Publishing AG, part of Springer Nature Archiv der Mathematik https://doi.org/10.1007/s00013-018-1198-5 Transfer of quadratic forms and of quaternion algebras over quadratic field extensions Karim Johannes Becher, Nicolas Grenier-Boley, and Jean-Pierre Tignol Abstract. Two different proofs are given showing that a quaternion al- gebra Q defined over a quadratic ´ etale extension K of a given field has a corestriction that is not a division algebra if and only if Q contains a quadratic algebra that is linearly disjoint from K. This is known in the case of a quadratic field extension in characteristic different from two. In the case where K is split, the statement recovers a well-known result on biquaternion algebras due to Albert and Draxl. Mathematics Subject Classification. 11E04, 11E81, 12G05, 16H05. Keywords. Isotropy, Witt index, Corestriction, Albert form, Character- istic two. 1. Introduction. A well-known theorem of Albert states that if a tensor prod- uct of two quaternion division algebras Q , Q over a field F of characteristic 1 2 different from 2 is not a division algebra, then there exists a quadratic ex- tension L of F that embeds as a subfield in Q and in Q ; see [6,

Journal

Archiv der MathematikSpringer Journals

Published: May 30, 2018

References

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