Central polynomials for matrices over finite fields

摘要

Let c(x _1,., x _d) be a multihomogeneous central polynomial for the n × 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{double-struck}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.