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.
Highlights
For a fixed positive integer k, we consider R Fkk[D] F[D] kk Let L be the field extension of Fofdegree k and a be the F-automorphism of L given by a(a) aq, a E L We fix a normal basis
We end this paper with a brief outline of an application of the cr-invariant sequences to recurring planes A recurring plane over a Galois field F is a matrix,4 [a3] over F, indexed by the set of natural numbers and for which there exist positive integers p, q satisfying a3 a+p. a,:l+q for all i, j Any such ordered pair (p, q) is called a period of the plane Any consecutive k rows of A constitute a matrix A [a], s _< _< k + s 1, j _> 0
J’(D) such that (f(D)) is cr-invariant, each s f(f(D)) gives a row(J’(D))-plane A [a,] whose i-th row equals an s-th row of S if s(mod k) The set of these planes can be seen to be closed under component-wise addition, shifts of rows, and of columns Their detailed study will be done in some later paper
Summary
Let F be a Galois field of order q and R Fkk[D], for a fixed positive integer k The set I’k(F). Of all sequences over F a,D R, a Fk, f(D)S- (s’) with s’ asn+, [3] For any regular f(D) R, the set fk(f(D)) {S Fk(F):f(D)S =0} is a finite F [D] -module, whose members are periodic sequences Let L be the field extension of F of degree k Fix a normal basis. ’ r/: R R such thatA M-1AM, A R k(f(D)) is said to be cr-invariant (or invariant under the Galois group G.L/F)) if for any S (sn) f(f(D)), S (cr(s,)) k(f(D)) A brief outline of an application of a cr-invariant fk(f(D)) to the construction of recurring planes is given at the end of this paper Given a regular f(D) E R, if f’ (D) f(D) or f (D) is a left circulant matrix, f(f(D)) is a-invariant Here we consider the converse in the sense that if(f(D)) is -invariant, does there exist a 9(D) R such that 9’(D) 9(D) and (f(D)) f(9(D)) In this paper we give a complete answer for the case/ 2, in Theorems (2) and (3) We give an explicit construction of a generating set and the dimension of an f2 (f(D)) if f’(D) f(D), in Theorem 4 An illustration of Theorem 4 is given in Example 15 The case, for any k: > 3 remains unsolved
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