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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Cyclotomic splitting fields

Author: Murray M. Schacher
Journal: Proc. Amer. Math. Soc. 25 (1970), 630-633
MSC: Primary 12.10
MathSciNet review: 0257037
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Abstract: Suppose $k$ is an algebraic number field and $D$ a finite-dimensional central division algebra over $k$. It is well known that $D$ has infinitely many maximal subfields which are cyclic extensions of $k$. From the point of view of group representations, however, the natural splitting fields are the cyclotomic ones. Accordingly it has been conjectured that $D$ must have a cyclotomic splitting field which contains a maximal subfield. The aim of this paper is to show that the conjucture is false; we will construct a counter-example of exponent $p$, one for every prime $p$.

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Keywords: Cyclotomic, local invariant, maximal subfield, norm, splitting field, totally ramified, valuation
Article copyright: © Copyright 1970 American Mathematical Society