Minimal cyclotomic splitting fields for group characters

Abstract

Let F F be a finite Galois extension of the rational number field Q Q , and let G G be a finite group of exponent n n with absolutely irreducible character χ \chi . This paper provides sufficient conditions for the existence of a minimal degree splitting field L L with F ( χ ) ⊆ L ⊆ F ( ε n ) F\left ( \chi \right ) \subseteq L \subseteq F\left ( {{\varepsilon _n}} \right ) , where ε n {\varepsilon _n} is a primitive n 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 L as above which is cyclic over F ( χ ) F\left ( \chi \right ) . The machinery we use to achieve the above results are certain genus numbers of F ( χ ) F\left ( \chi \right ) .