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Proceedings of the American Mathematical Society

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



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