Field generated by character values

Definition

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