All dihedral division algebras of degree five are cyclic
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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.References
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Additional Information
- Eliyahu Matzri
- Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan, 52900, Israel
- Email: elimatzri@gmail.com
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 1925-1931
- MSC (2000): Primary 16K20, 12E15
- DOI: https://doi.org/10.1090/S0002-9939-08-09310-6
- MathSciNet review: 2383498