Field generated by character values is splitting field implies it is the unique minimal splitting field
From Groupprops
Statement
Let be a finite group. Consider a characteristic for fields that is either zero or a prime not dividing the order of
. Suppose
is the field generated by the values of all characters of linear representations of
in that characteristic.
Then, if is a Splitting field (?) for
(i.e., all linear representations of
in that characteristic can be realized over
), it is a Minimal splitting field (?) for
and is the unique minimal splitting field up to isomorphism.
Note that if all the linear representations of in that characteristic have Schur index 1, then
is indeed a splitting field. However, it is possible for
to be a splitting field even if some of the representations have Schur index greater than 1.