Field generated by character values is splitting field implies it is the unique minimal splitting field

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Statement

Let G be a finite group. Consider a characteristic for fields that is either zero or a prime not dividing the order of G. Suppose K is the field generated by the values of all characters of linear representations of G in that characteristic.

Then, if K is a Splitting field (?) for G (i.e., all linear representations of G in that characteristic can be realized over K), it is a Minimal splitting field (?) for G and is the unique minimal splitting field up to isomorphism.

Note that if all the linear representations of G in that characteristic have Schur index 1, then K is indeed a splitting field. However, it is possible for K to be a splitting field even if some of the representations have Schur index greater than 1.

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