Quadratic fields of values of finite solvable groups

Abstract

We show that if $G$ is a finite solvable group such that $E = \mathbb Q(\{\chi(g)\, |\, \chi\in{\rm Irr}(G),\ g \in G\})$ is an extension of degree 2 over the rationals, then $E = \mathbb Q(\sqrt{\pm d})$ for a positive integer $d$ all of whose prime divisors lie in $\{2, 3, 5\}$. Moreover, if $G$ has a composition factor of order 7, then either $E = \mathbb Q(\sqrt{2})$ or $E = \mathbb Q(\sqrt{-3})$. This provides a partial answer to a question of G. Navarro.