Let $F$ be a field and let $G$ be a group. Let $V$ be an irreducible $F$-linear representation of $G$.
If $F$ is algebraically closed, then $\dim_F \,(\operatorname{Hom}_G(V,V)) = 1$ by Schur's lemma.
I would like to know if there is a way to calculate $\dim_F \,(\operatorname{Hom}_G(V,V))$ when $F$ is not necessarily algebraically closed. If $V$ is absolutely irreducible, then Schur's lemma gives the answer, but I'm not sure what happens when this is not the case.