Field generated by character values need not be a splitting field

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Statement

In characteristic zero

It is possible to have a finite group such that the Field generated by character values (?) in characteristic zero is not a Splitting field (?).

Related facts

Proof

Further information: quaternion group, faithful irreducible representation of quaternion group, linear representation theory of quaternion group

The idea is to pick a group where one or more of the representations have Schur index greater than 1. The smallest example is the quaternion group, where the field generated by character values is the field of rational numbers (so the group is a rational group) but the two-dimensional faithful irreducible representation cannot be realized over the rationals or even over the reals (this can be shown via the indicator theorem).