INVARIANCE OF RECURRENCE SEQUENCES UNDER A GALOIS GROUP

Let F be a Galois field of order q, k a fixed positive integer and RF [D] where D is an indeterminate Let L be a field extension of F of degree k: We identi LF with F via a fixed normal basis B of L over F The F-vector space Fk(F) F(L)) of all sequences over F is alett R-module For any regular f(D) E R, fk(f(D)) {S I’k(F) f(D)S 0} is a finite F[D]module whose members are ultimately periodic sequences The question of invariance of a ftk(f(D)) under the Galois group G of L over F is investigated


INTRODUCTION.
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 ultimately periodic sequences Let L be the field extension of F of degree k Fix a normal basis k-I B {a, aq, a q2 a q-I } of L over F such that ' a q' 1 Through this basis we identify Lv with !=0 F kxl The Galois group G(L/F) is generated by a L L such that or(a) aq, a L The matrix of z relative to B is the companion matrix M of X k 1 We get the inner automorphism r/: R R such thatA ' M-1AM, A R Then 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, then 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 also 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 2.

PRELIMINARIES
Let F be a Galois field of order q and F(F) be a lef F [D] -module of all sequences over F, [2]  For any f(D) =/= 0 in F[D], isomorphic to F[D]/F[D]f(D) For any two non-zero polynomials f(D), 9(D) F[D], f(D) A 9(D) and f(D) V 9(D) will denote their gcd and lcm respectively, 0 A f(D) is the monic factor of f(D) of degree same as deg f(D) The following is well known (see [1] or [2]) THEOREM 1.For any two non-zero polynomials f(D), g(D 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 k-! B {cr, aq, a q } of L over F satis,ing ' a q' 1 By using this we identifi] L with F Then I--0 Homt..(L, L) F and a is given by the k x k-matrix 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 the companion matrix of X k 1 Then where b, a,+l 3+1, -]-1, j + 1 are positive integers modulo k.The following is immediate LEMMA 1.For A [a,3] R, M -1 AM A iff A al 0,2 akak ak al ak-2 ak-1 a,'2 a3 ak al for some a, F[D].
For any A R, A ' denotes M-IAM If f(D) R is regular, then the bound of f(D) is the smallest degree monic polynomial d(D) [3].Further fk(f(D)) f'(D)fk(d(D)Ik), RFtk(f(D)) gt(d(D)I) and g&(d(D)I) f(d(D))1, [3].For any module N, N denotes the direct sum of k copies of N.

3.
A a-INVARIANT 12k(f(D)) We start with the following LEMMA 2. Let f(D), g(D) R, both be regular Rf(D) Rg(D).
PROPOSIT|ON I.For any regular f(D) R, the following are equivalent The above proposition shows that if Rf(D)= Rg(D) for some g(D) R satisfying gn(D) g(D), then (f(D)) is cr-invariant Is the converse true9 We investigate this question LEMMA 3. Let f(D) R be regular such that Rf(D) Rf'(D), let f(D) Xf'(D) The following hold" (i) det(X) 1 (ii) There exists g(D) R such that 9'(D) g(D) and Rf(D) Rg(D) iff for some invertible A R, A' AX PROOF.(i) is obvious Let g(D) exist, then g(D) Af(D) for some invertible A R Then g(D) g'(D), gives AXfO(D) A'f'(D) Hence A AX.The converse is obvious.LEMMA 4. Let f(D) and X be as in Lemma 3 Let X be obtained from X by applying the cyclic permutation ,k (1,2, 3, k) to the columns of X Then some k-th root of unity, in some field extension of F, is a characteristic value of Xa.
u=l where u + 1, j + 1 are least positive residues modulo k This is a system of k homogeneous linear equations in a.By arranging a.'s in the order all, a21, akl, a12, a22, ak2, we get the coefficient matrix, the k x k--matrix We now solve for all, a22, such that A R and det(A) 1 Then (3 2) gives 2alia22 aal ca2 52 By taking a 0 in F, this equation gives a._,2 E F[D].Similarly if a 0, we can solve for al and a2 Let a 0 :/: c By multiplying (3 3)  As c divided 1-b" (1-b)(1 + b), and 1-b, 1 + b are coprime, write c CLC2, with ct and c2 factors of 1 + b and 1 b respectively Put a-d=c, a+d=c2.
To be definite, take Y all bd So that 1 1 (1 + b) We now consider the case of char F 2 (II) be0, at least one of a and c is non-zero, a/Xc=l, a=r and c=s for some r, ( Asac+b =l, aAb=bAc=l Then (3 6) yieldsaAc=l Further (3 6) yields This immediately yields a r for some r R Similarly c s for some s 6 R 1 0] is a solution Let (II) hold.Then r A s 1 So for Conversely if (I) holds, A 0 a some x,yR rx + sy b.
Theorem 4 gives that (A) consists of all sequences of least periods, factors of 8, with first eight terms 2c+d 2b+d 2a+c+d b+c+d a+b+2c+d a+2b+c+d 2a+b+d 2a+b+d a+d with a,b,c,dF.
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. Each column of A being a member of F k 1, we can regard A', a sequence in r(F) Given a regular J'(D) E Fkk[D], call a recurring plane a row(f(D))-plane, if every submatrix of A constituted by any k consecutive rows of A, is a member of fk(f(D)).Given an 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 easily seen to be closed under component-wise addition, shifts of rows, and of columns Their detailed study will be done in some later paper Then fk(f(D)) k(g(D)) iff PROOF.Let d(D) bound (f(D)), d'(D) bound (g(D)) Let a sequence S E F(F) be a generator of the F [D] -module f2(d(D)) By [3, Lemma (2 4)], the mapping " R/Rd(D) fl(d(D))' [Q, (d(D)L,)] such that for any [g,(D)]

THEOREM 2 .
the k k-identity matrix As I and X commute, C as a matrix over F[X'X,I] c_ F'[D], has determinant !(X)k So for some matrix C' over F[X', 1],CC' diagkk[I-(XX)k, ..,I-(XX)].By taking determinant over F[D], we get det(C)det(C') [det(I-(xA)k)] k As C is singular, we get det(I-(XX)k) 0. This completes the proof a b with COROLLARY 1.For k 2, under the hypothesis of Lemma 4, X--b c ac+b 2-1PROOF.Now X x x2 x As 1 or 1 is a characteristic value of Xa, and by Lemma Let F be a Galois field of characteristic p 2 Ifa regular f(D) E R F22[D]   is such that 122(f(D)) is invariant under a, then f2(f(D)) f2(g(D)) for some g(D) ac + b 1 In view of Lemma 3 we find an A I_ a21 a22 R with 0 :/= det(A) F such that A n AX, b -fi 0 By solving the system of linear equations it can be seen that ) b-212alla22 aal ca2 ( by c, and by putting Y ca22, we get ,,_,2 shouldbe suchthat ai -c forsomed E F[D] Then (all d)(all + d) c.

TItEOREM 3 .
Let  F be a Galois field of characteristic 2 Let f(D) R F22[D] tt satisfying Rf(D) Rg(D) and f(D) g(D) iffone ofthe following holds (I) b 0

2 a
s e F[D].PROOF.Let Rf(D) Rg(D) with f(D) g(D) By Lemma 3 we get an invertible A in R such that A n AX Let b 7 0 As in the proof of Theorem 2 b a +b ca.2.