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