Cyclotomic character fields and sets of primes

Abstract

Let $\ddot{I}€=\{2,q\}$ where $q$ is an odd prime. Let $G$ be a finite group of order divisible by a prime $p\in \ddot{I}€$. We show that the principal $p$-block of $G$ contains a nontrivial irreducible character of degree not divisible by $2$ nor $q$ and with field of values contained in the $q$th cyclotomic extension. This statement simultaneously provides a principal block version of results of Navarro--Tiep and Giannelli--Hung--Schaeffer Fry--Vallejo.