# Field generated by character values

From Groupprops

## Contents

## Definition

Suppose is a finite group. Pick a characteristic that is either zero or a prime not dividing the order of . The **field generated by character values** for in that characteristic is the smallest field in that characteristic containing the values of all the characters of irreducible representations of over a splitting field in that characteristic.

## Facts

### Relationship with cyclotomic extensions

- In characteristic zero, field generated by character values is contained in a cyclotomic extension of rationals, because characters are cyclotomic integers.
- Field generated by character values need not be cyclotomic

### Uniqueness and relationship with splitting fields

- The field generated by character values is unique up to isomorphism of fields.
- The field generated by character values is contained in every splitting field, and hence also in every minimal splitting field.
- Field generated by character values is splitting field implies it is the unique minimal splitting field
- Field generated by character values need not be a splitting field