  Let $c(x_1,...,x_d)$ be a multihomogeneous central polynomial for the
$n\times n$ matrix algebra $M_n(K)$ over an infinite field $K$ of positive
characteristic $p$. We show that there exists a multihomogeneous polynomial
$c_0(x_1,...,x_d)$ of the same degree and with coefficients in the prime field
$F_p$ which is central for the algebra $M_n(F)$ for any (possibly finite) field
$F$ of characteristic $p$. The proof is elementary and uses standard
combinatorial techniques only.

Rings and Algebras