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Cyclotomic Splitting Fields

Published online by Cambridge University Press:  20 November 2018

R. A. Mollin
R. A. Mollin*
Affiliation:
Department of Mathematics And Statistics Queen's University, Kingston, OntarioK7L 3N6
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Abstract

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Let D be a division algebra whose class [D] is in B(K), the Brauer group of an algebraic number field K. If [D⊗KL] is the trivial class in B(L), then we say that L is a splitting field for D or L splits D. The splitting fields in D of smallest dimension are the maximal subfields of D. Although there are infinitely many maximal subfields of D which are cyclic extensions of K; from the perspective of the Schur Subgroup S(K) of B(K) the natural splitting fields are the cyclotomic ones. In (Cyclotomic Splitting Fields, Proc. Amer. Math. Soc. 25 (1970), 630-633) there are errors which have led to the main result of this paper, namely to provide necessary and sufficient conditions for (D) in S(K) to have a maximal subfield which is a cyclic cyclotomic extension of K, a finite abelian extension of Q. A similar result is provided for quaternion division algebras in B(K).

Keywords

16A2616A65Cyclotomic fieldsplitting fielddivision algebramaximal subfield
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 25 , Issue 2 , 01 June 1982 , pp. 222 - 229
Copyright
Copyright © Canadian Mathematical Society 1982

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