Uniformly semi-rational groups

TL;DR

This paper introduces uniformly semi-rational groups, exploring their character values on quadratic extensions, and classifies related invariants for finite nilpotent groups, advancing understanding of their algebraic properties.

Contribution

It defines uniformly semi-rational groups, introduces invariants for finite groups, and classifies possible values for nilpotent groups' invariants and character value fields.

Findings

Invariants for finite groups are characterized.

Classification of fields generated by character values.

Determination of invariants for finite nilpotent groups.

Abstract

We introduce and study some families of groups whose irreducible characters take values on quadratic extensions of the rationals. We focus mostly on a generalization of inverse semi-rational groups, which we call uniformly semi-rational groups. Moreover, we associate to every finite group two invariants, called rationality and semi-rationality of the group. They measure respectively how far a group is from being rational and how much uniformly rational it is. We determine the possible values that these invariants may take for finite nilpotent groups. We also classify the fields that can occur as the field generated by the character values of a finite nilpotent group.