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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

All dihedral division algebras of degree five are cyclic


Author: Eliyahu Matzri
Journal: Proc. Amer. Math. Soc. 136 (2008), 1925-1931
MSC (2000): Primary 16K20, 12E15
Published electronically: February 7, 2008
MathSciNet review: 2383498
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Abstract: In 1982 Rowen and Saltman proved that every division algebra which is split by a dihedral extension of degree $ 2n$ of the center, $ n$ odd, is in fact cyclic. The proof requires roots of unity of order $ n$ in the center. We show that for $ n=5$, this assumption can be removed. It then follows that $ {}_{5\:}\operatorname{Br}(F)$, the $ 5$-torsion part of the Brauer group, is generated by cyclic algebras, generalizing a result of Merkurjev (1983) on the $ 2$ and $ 3$ torsion parts.


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Additional Information

Eliyahu Matzri
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan, 52900, Israel
Email: elimatzri@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09310-6
PII: S 0002-9939(08)09310-6
Keywords: Central simple algebras, cyclic algebras
Received by editor(s): November 27, 2006
Published electronically: February 7, 2008
Additional Notes: The author thanks his supervisors, L. H. Rowen and U. Vishne, for many interesting and motivating talks and for supporting this work through BSF grant no. 2004-083.
Communicated by: Martin Lorenz
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.