All dihedral division algebras of degree five are cyclic

Eliyahu Matzri

doi:10.1090/S0002-9939-08-09310-6

All dihedral division algebras of degree five are cyclic

Eliyahu Matzri

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3 Scopus citations

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 ₅Br(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.

Original language	English
Pages (from-to)	1925-1931
Number of pages	7
Journal	Proceedings of the American Mathematical Society
Volume	136
Issue number	6
DOIs	https://doi.org/10.1090/S0002-9939-08-09310-6
State	Published - 1 Jun 2008
Externally published	Yes

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1090/S0002-9939-08-09310-6

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title = "All dihedral division algebras of degree five are cyclic",

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 5Br(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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AB - 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 5Br(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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All dihedral division algebras of degree five are cyclic

Abstract

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