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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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Eliyahu Matzri
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan, 52900, Israel

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.

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