Invariance of recurrence sequences under a galois group

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