Minimal cyclotomic splitting fields for group characters

Mollin, R. A.

doi:10.1090/S0002-9939-1984-0744629-0

Minimal cyclotomic splitting fields for group characters
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Proc. Amer. Math. Soc. 91 (1984), 359-363 Request permission

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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