Primes and The Field of Values of Characters

TL;DR

This paper explores the classification of fields of values of complex irreducible characters of finite groups, extending existing conjectures to characters with degrees divisible by powers of primes, and provides supporting evidence.

Contribution

It generalizes the classification conjecture to include characters with degrees divisible by arbitrary powers of primes, advancing understanding in character theory.

Findings

Extended the conjecture to characters with degrees divisible by powers of p

Provided evidence supporting the extended conjecture

Connected classification of fields of values with prime divisibility properties

Abstract

Let $p$ be a prime. For $p=2$, the fields of values of the complex irreducible characters of finite groups whose degrees are not divisible by $p$ have been classified; for odd primes $p$, a conjectural classification has been proposed. In this work, we extend this conjecture to characters whose degrees are divisible by arbitrary powers of $p$, and we provide some evidence supporting its validity.