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

Abstract

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 $\mathbb{Q}(\chi (g))=\mathbb{Q}(\chi )$? The answer was recently shown to be no in general by Ulrich Thiel. However, when $\chi$ is further assumed to be factorizable as a product of $p$-special characters, we are able to show that the answer is yes.

arXiv comment12 pages, strengthened Theorem A, removed Theorems B and C, removed Section 3