Central polynomials for matrices over finite fields

Abstract

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 _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.

Keywords:

• central polynomials
• matrix algebras

AMS Subject Classifications::

• 16R10
• 16R30

Acknowledgements

The research of the first named author was partially supported by the Slovenian Research Agency (Program No. P1-0288).