On a question of Navarro on the field of values of characters of solvable groups

RESUMO

In a 2023 survey paper, Gabriel Navarro posed the following problem: Given an irreducible character $\chi$ of some solvable group $G$ where $\chi$ either is 2-rational or has odd degree, does there exist some $g \in G$ such that $\mathbb{Q}(\chi(g)) = \mathbb{Q}(\chi)$? When $\chi$ is imprimitive, the answer is no in general. In the case where $\chi$ is primitive (or even factorizable as a product of $p$-special characters), we are able to show that there exists $g \in G$ such that $\Gal(\mathbb{Q}(\chi)/\mathbb{Q}(\chi(g)))$ is an elementary abelian 2-group. We are then able to show that the answer to Navarro's problem is yes when we further assume that either $\chi$ is 2-rational, $c(\chi)$ is divisible by at most two primes, or $\chi(1)$ is a power of an odd prime.