Showing that End$_{FG}V$ is isomorphic to $F$ as a ring.

Question

Let $G$ be a finite group, $F$ be an algebraically closed field, and $V$ an irreducible $FG$-module. I want to show that End$_{FG}V$ is isomorphic to $F$ as a ring.

I managed to get that End$_{FG}V$ is a division ring by Shur's lemma, and that under composition End$_{FG}V$ is a field, but I am struggling to finish the proof off. I think that there may be something to do with $F$ being algebraically closed that I am missing.

Answer 1

Since $V$ is simple (irreducible), it is finite-dimensional over $F$. Then $\mathop{\rm End}_{FG}(V)$ is a finite-dimensional division algebra over $F$. But $F$ being algebraically closed implies that any such algebra is $1$-dimensional, and thus isomorphic to $F$ itself.