Invariance of recurrence sequences under a Galois group (Q1912330)

Let \(F\) be a Galois field of order \(q,k\) a fixed positive integer and \(R= F^{k\times k} [D]\) for \(D\) an indeterminate. Let \(L\) be a field extension of \(F\) of degree \(k\) with fixed basis \(B=\{\alpha, \alpha^q, \dots, \alpha^{q^{k-1}}\}\) and identify \(L_F\) with \(F^{k \times 1}\). The \(F\)-vector space \(\Gamma_k(F)\) of all sequences over \(F^{k \times 1}\) is a left \(R\)-module. For any regular \(f(D)\in R\), \(\Omega_k(f(D)) = \{S\in\Gamma_k(F) : f(D)S = 0\}\) is a finite \(F[D]\)-module. The Galois group \(G(L/F)\) is generated by \(\sigma(a) = a^q\), \(a\in L\). The question of the invariance of an \(\Omega_k(f(D))\) under the Galois group is considered. A complete answer is given for the case \(k=2\) and an explicit construction of a generating set and the dimension of an \(\Omega_2(f(D))\) is given if \(f^\eta(D) = f(D)\) for \(\eta\) an inner automorphism of \(R\). The case for \(k>3\) remains unsolved.