Groups with small multiplicities of fields of values of irreducible characters

TL;DR

This paper classifies finite groups where each field extension of the rationals is the field of values for at most three irreducible characters, advancing understanding of the relationship between group structure and character fields.

Contribution

It provides a complete classification of such finite groups, a novel result connecting group theory and field extension properties of characters.

Findings

Identifies all finite groups with the specified property.

Establishes bounds on the number of irreducible characters with a given field extension.

Enhances understanding of the interplay between group structure and character fields.

Abstract

In this work, we classify all finite groups such that for every field extension F of \mathbb{Q}, F is the field of values of at most 3 irreducible characters.