# Irreducible representation over splitting field surjects to matrix ring

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## Statement

Suppose $G$ is a finite group and $k$ is a splitting field for $G$. Suppose $\rho:G \to GL(n,k)$ is an irreducible representation for $G$ over $k$. Then, the map extends uniquely by $k$-linearity to a $k$-linear map from the group ring to the matrix ring:

$\tilde{\rho}: kG \to M(n,k)$

The claim is that $\tilde{\rho}$ is surjective.

Instead of requiring $k$ to be a splitting field, we can require only that $k$ have characteristic not dividing the order of $G$ and the representation $\rho$ be absolutely irreducible.