# 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.