Let $A$ and $B$ be $n\times n$ matrices over the Galois Field of order $p$ ($p$ is a prime). Suppose that $A$ and $B$ are diagonizable matrices and that they commutate. Is it possible to make them simultaneously diagonizable in $GF(p)$?
I know that when we have an algebraically closed field the things go fine. What in this case? I find somethings that lead me to think that this is true.