Extensions of Schönemann's theorem in Galois rings

Abstract

Let p be a prime and Fp be a finite field of p elements. For a1,...,ak,b∈Fp, let NFp(a1,...,an,b) denote the number of the solutions (x1,...,xn)∈Fpn to the following linear equationsa1x1+⋅⋅⋅+anxn=b with the constraint xi≠xj for all i≠j. In 1839, Schönemann obtained the exact formula of NFp(a1,...,an,0) provided that ∑i=1nai=0 and ∑i∈Iai≠0 for all I⫋{1,...,n}. Schönemann's theorem was generalized by Bibak, Kapron and Srinivasan from Fp to Z/mZ for any positive integer m. Recently, Li and Yu generalized Schönemann's theorem to the finite field Fq with general prime power q. In this article, we give a further extension of Schönemann's theorem in the Galois rings GR(p2,r).