# Field generated by character values

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## 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

- 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.
- The field generated by character values is a cyclotomic extension of the rationals (in characteristic zero) because characters are cyclotomic integers.
- 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