Invariance of recurrence sequences under a galois group

Abstract

Let F be a Galois field of order q, k a fixed positive integer and R=Fk×k[D] where D is an indeterminate. Let L be a field extension of F of degree k. We identify Lf with fk×1 via a fixed normal basis B of L over F. The F-vector space Γk(F)(=Γ(L)) of all sequences over Fk×1 is a left R-module. For any regular f(D)∈R, Ωk(f(D))={S∈Γk(F):f(D)S=0} is a finite F[D]-module whose members are ultimately periodic sequences. The question of invariance of a Ωk(f(D)) under the Galois group G of L over F is investigated.