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 fact about::splitting field for $$G$$ (i.e., all linear representations of $$G$$ in that characteristic can be realized over $$K$$), it is a fact about::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