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Minimal cyclotomic splitting fields for group characters


Author: R. A. Mollin
Journal: Proc. Amer. Math. Soc. 91 (1984), 359-363
MSC: Primary 11R18; Secondary 20C05
DOI: https://doi.org/10.1090/S0002-9939-1984-0744629-0
MathSciNet review: 744629
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Abstract: Let $ F$ be a finite Galois extension of the rational number field $ Q$, and let $ G$ be a finite group of exponent $ n$ with absolutely irreducible character $ \chi $. This paper provides sufficient conditions for the existence of a minimal degree splitting field $ L$ with $ F\left( \chi \right) \subseteq L \subseteq F\left( {{\varepsilon _n}} \right)$, where $ {\varepsilon _n}$ is a primitive $ n$th root of unity. We obtain as immediate corollaries known results pertaining to this question in the literature. Moreover we obtain necessary and sufficient conditions for the existence of a minimal splitting field $ L$ as above which is cyclic over $ F\left( \chi \right)$. The machinery we use to achieve the above results are certain genus numbers of $ F\left( \chi \right)$.


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DOI: https://doi.org/10.1090/S0002-9939-1984-0744629-0
Article copyright: © Copyright 1984 American Mathematical Society

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