Irreducible representation over splitting field surjects to matrix ring
Suppose is a finite group and is a splitting field for . Suppose is an irreducible representation for over . Then, the map extends uniquely by -linearity to a -linear map from the group ring to the matrix ring:
The claim is that is surjective.
Instead of requiring to be a splitting field, we can require only that have characteristic not dividing the order of and the representation be absolutely irreducible.