# On diagonizability of commutating matrices

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.

Yes, this will work fine, as long as we assume that $A$ and $B$ have all their eigenvalues in $GF(p)$ (or, equivalently, strengthen your assumption to say that $A$ and $B$ are diagonalizable over $GF(p)$. Possibly this is already what you intended.) The proof which you cite works over any field; there's no need for it to be algebraically closed.