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

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.

Related facts

• Field generated by character values need not be a splitting field
• Minimal splitting field need not be unique
• Minimal splitting field need not be cyclotomic
• Sufficiently large implies splitting
• Splitting not implies sufficiently large