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Finite dimensional factor algebras of F Miroslav Kureš 2[X1, …, Xn] and their fixed point subalgebras Institute of Mathematics Brno University of Technology Brno, Czech Republic kures@fme.vutbr.cz Abstract—Fixed point subalgebras of finite dimensional factor algebras of algebras of polynomials in n indeterminates over the finite field F 2(with respect to all F Keywords-polynomial; finite field; group of automorphisms; fixed point 2-algebra automorphisms) are fully described. 1. Introduction In [6], we consider local commutative R -algebra Awith identity, the nilpotent ideal nAof which has a finite dimension as a vector space and A / nA= R and study its subalgebra SA of fixed elements, SA ={aA; M (a) = a for all M Aut R A}, where Aut R Ais the group of R -automorphisms of the algebra A. This research is motivated by differential geometry, wherealgebras in question are usually called Weil algebras and, in particular, the bijection between all natural operators lifting vector fields from m-dimensional manifolds to bundles of Weil contact elements and the subalgebra of fixed points SA of a Weil algebra Awas determined (in [4]). Although in the known geometrically motivated examples is usually SA = R (such SA is called trivial), there are some algebras for which SA is a proper superset of R In this paper, we simply replace and they call attention to the geometry of corresponding bundles. Thus, the fundamental problem is a classification of algebras having SA nontrivial. See [4], [5] for related geometric questions and the survey paper [8] for known results up to now, especially for a number of claims concerning the form of subalgebras of fixed points of various Weil algebras. R by F 2and study quite analogous questions. We come to a different situation by this: mainly, factor rings are finite 2. Polynomials over F2 rings (see [1]) and there is the whole theory about this topic. It is known the ring automorphism problem lying in a decision if a finite ring has a non-identical automorphism or not. Results about fixed point subalgebras are also qualitatively totally different from the real case and they can have interesting applications in the coding theory and cryptography. Polynomials in nindeterminates over F a: 2are the maps of the type N 0no F i.e. a multiindex maps onto an element of 2, F 2; the support of the map must be by definition finite. We define the addition of polynomials and the multiplication of polynomials by the usual way and use also the standard denotation for them. However, we consider only polynomials in this paper and not their evaluations (polynomial maps); the fatal inaccuracy of such a confusion is explained e.g. in [7]. With the mentioned operations, polynomials over F 2form the ring denoted by F 2[X1, …, Xn] or shortly by F II.1. Ideals in F 2[X] 2[X]. The (unique) maximal ideal of F 2[X] is m = (X1, …, Xn). Powers of m represent notable class of ideals. We mention another important ideal. For finite fields F q, the field ideal in F q[X1,…,Xn] isdefined as f = (X1q –X1, …, Xn q –Xn). Thus, we have f = (X12+X1, …, Xn 2+Xn). for q= 2. 3. ALGEBRAS ( D 2)n r In this section, we will study factor rings ( D 2)n r = F 2[X1, …, Xn] / mr+1, where r N . III.1. Dual numbers over F 2 As D = R [X] / (X2) is usually called the algebra of dual numbers(which is definable promptly by D ={a0+a1X; a0, a1 R , X2= 0}), we obtain forr = n = 1 D 2= ( D 2)11= F 2[X] / (X2) the algebra of dual numbers over F 2. Elements of D 2are ex- pressible in the form Published results were acquired using the subsidization of the GA N o. 201/09/0981. Open Journal of Applied Sciences Supplement：2012 world Congress on Engineering and Technology 212 Cop y ri g ht © 2012 SciRes. a0+a1X;a0, a1 F 2,X2= 0. We observe that D 2has the following additive and multiplicative tables: + 0 1 X 1 +X 0 1 X 1 +X 0 1 X 1 +X 1 0 1 + X X X 1 + X 0 1 1 + X X 1 0 × 0 1 X 1 +X 0 1 X 1 + X 0 0 0 0 0 1 X 1 + X 0 X 0 X 0 1 + X X 1 As to classification of B. Fine, [2], this finite ring can be expressed as (a,b; 2a= 2b= 0, a2= 0, b2=b, ab =a, ba =a). (the case denoted by 'G' in [2]). Furthermore, in [3] is presented that dual numbers over F 2, complex numbers over F 2and paracomplex numbers over F 2 are isomorphic rings.We find easily the following result. (Analogously to the real case, by the subalgebra of fixed elementsSA of an F 2-algebra Awe mean the subalgebra of all elements asatisfying M (a) = a for all F 2-automorphisms M of A and if SA= F 2, we call SA trivial.) PROPOSITION 1. The group of all F 2-automorphisms of D 2is trivial. It follows that for A = D 2is SA nontrivial as SA = A. PROOF. Let M be an F 2-automorphism of D 2. As M (1) = 1, M is fully determined by a specification of M (X). In general, M (X) = b0+b1X;b0, b1 F 2. However, we have 0 = M (0) = M (X2) = M (X) M (X) = b02+b12X2=b02; thus, b0= 0, then, necessarily, b1= 1 for M be a bijection. So, the group of all F 2-automorphisms of D 2contains only one element: the identical automorphism. Then the rest of the claim becomes evident. III.2. The case r>1, n=1 Elements of the algebra ( D 2)r= ( D 2)1r= F 2[X] / mr+1 = F 2[X] / (Xr+1 ) have a form a0+a1X + a2X2+…+ arXr;a0, a1, a2, …, ar F 2, Xr+1 = 0. We start with the following lemma. LEMMA 1. Every endomorphism M : ( D 2)ro( D 2)rdeter- mined by M (1) = 1 M (X) = X+b2X2+…+brXr;b2, …, br F 2 belongs to the group of all F 2-automorphisms of ( D 2)r. PROOF.It suffices to describe M : we have Y = M (X) = X+b2X2+ … + brX Y2=X2+ terms of degree > 2 … Yr-1 =X r-1+terms of degree > r-1 Yr=Xr The last equation provides Xr by Y’s, the last but one provides (after the substitution) Xr-1and so on. We recall that nAdenotes the ideal of nilpotent elements of A(nilradical of A). If an element aAhas the property au = 0 for all unA, we call athe socle element of A. It is easy to find that all socle elements constitute an ideal; this ideal is called the socle of Aand denoted by soc(A). Now, we can formulate the main result about automorphisms of ( D 2)rincluding also the case r = 1. PROPOSITION 2. For r N , let A= ( D 2)r.Every automor- phism M : AoA has a form M (1) = 1 M (X) = X+b2X2+ … + brXr;b2, …, br F 2. It follows SA is always nontrivial, in particular soc(A)SA. PROOF. It is evident that the endomorphism M (1) = 1 M (X) = b2X2+ … + brXr;b2, …, br F 2 does not represent an automorphism. Further, soc(( D 2)r )= {aXr;a F 2} and M (Xr) = Xrwas demonstrated already in the previous lem- ma. III.3. The case r=1, n>1 Elements of the algebra ( D 2 )n= ( D 2)n 1= F 2[X1, …, Xn] / m2 = F 2[X1, …, Xn] / (X1, …, Xn)2have a form a0+a1X1+a2X2+…+anXn;a0, a1, a2, …, an F 2,XiXj= 0 for all i, j{1, …, n}. PROPOSITION 3. The group of all F 2-automorphisms of ( D 2)n is isomorphic to the general linear group GL(n,2) = GL(n, F 2) of the order n over F 2. PROOF. A general form of endomorphisms of ( D 2)nis M (1) = 1 M (X1) = b10+b11X1+b12X2+ … + b1nXn M (X2) = b20+b21X1+b22X2+ … + b2nXn … M (Xn) = bn0+bn1X1+bn2X2+ … + bnnXn. However, we have 0 = M (0) = M (X12) = M (X1) M (X1) = b102+b112X12+ … + b1n 2Xn 2=b102, thus, b10 = 0, and analogously, b20 = … = bn0= 0. Now, the matrix (bij) must be invertible for M be a bijection. So, Cop y ri g ht © 2012 SciRes.213 automorphisms of ( D 2)ncorrespondsexactly with the group GL(n,2). COROLLARY 1. Let i ,j^n}, izj.Then M (i,j):( D 2)no ( D 2)ngiven by M (i,j)(1) = 1 M (i,j)(Xi) = Xi+Xj M (i,j)(Xk) = Xkfor all k^ n}, kzi, belongs to the group of all F 2-automorphisms of ( D 2)n. PROOF.It is clear that M (i,j) meets the form from the proof of the previous proposition. REMARK 1.We remark that the order of GL(n, 2) is 3i=0n-1(2n-2i). PROPOSITION 4. For n N , n > 1, let A= ( D 2)n.Then the subalgebra SA of fixed points of A is always trivial. PROOF.First, we prove that the element X1+X2+… +Xn is not fixed. For this, it suffices to take some automorphism M (i,j), e.g. M (1,2)sends X1+X2+ … + Xnonto X1+X3+ … + Xn. Second, let {k1, …, kh}be a (non-empty) proper subset of ^n}, i.e. h < n. We provethat the element Xk1+Xk2+ … + Xkh is not fixed, too. We take i{k1, …, kh} and j^n} {k1,…,kh} and apply M (i,j): it sends Xk1+Xk2+ … + Xkhonto Xk1+Xk2+ … + Xkh +Xj . So, SA = F 2. III.4. The case r>1, n>1 Elements of the algebra ( D 2)n r= F 2[X1, …, Xn] / mr+1= F 2[X1, …, Xn] / (X1, …, Xn)r+1 have a form a0+ a1X1 +a2X2 + … + anXn + a11X12+a12 X1X2 + … + annXn 2+ …+ a1…1X1r+a1…12 X1r-1X2 + … + an…nXn r, a0,a1, …, an, a11, …, an…n F 2. On basis of previous results we can find out nature of this general case now. PROPOSITION 5. For r N ,n N, n>1,let A= ( D 2)n r.Then the subalgebra SA of fixed points of A is always trivial. PROOF. Obviously, elements of GL(n,2) represent automor- phisms also for ( D 2)nr.Of course, not all Let PAand let exist i,j^ n} such that wPwXi zand wPwXj Analogously with the case r= 1, n> 1, we apply M (i,j) for the demonstration that Pcannot be fixed. automorphisms, however, these (linear) automorphisms suffice for our following considerations. In the proof, we use formally partial derivations w wXjfor an expressing whether elements ofA contain Xjin some non-zero power or not. So, let QAis not of such a type and let Vbe a permutation of n-tuple (X1, …, Xn) for which V(Q)zQ. As permutations of (X1, …, Xn) are also elements of GL(n,2), we find again that Q cannot be fixed. Therefore we take RAsuch that wPwXizfor all i^ n} and such that does not exist any permutation of (X1, …, Xn) yielding a transformation of R. Nevertheless, a "symmetry" of Rwill be again unbalanced by M (i,j), e.g. M (1,2). Hence we have an automorphism for which not even Ris fixed. Thus, only zero degreeelements of Aremain fixed with respect to all automorphisms: SA is trivial. 3. Summary The previous assertions provide the following summary theorem (r,n N ). THEOREM.The subalgebra SA of fixed points of A = ( D 2)n r is nontrivial if and only if n = 1. REFERENCES [1] G. Bini and F. Flamini, Finite Commutative Rings and Their Applications, Kluwer Academic Publishers 2002. [2] B. Fine, “Classification of finite rings of order p2,” Mathematics Magazine 66, No. 4, 1993, pp. 248–252. [3] J.Hrdina, M. Kureš and P. Vašík, “A note on tame polynomial automorphisms and the security of TTM cryptosystem,” Applied and Computational Mathematics 9, No. 2, 2010, pp. 226–233. [4] M. Kureš and W. M. Mikulski, “Natural operators lifting vector fields to bundles of Weil contact elements,” Czechoslovak Mathematical Journal 54 (129), 2004, pp. 855–867. [5] M. Kureš and W. M. Mikulski, “Natural operators lifting 1-forms to bundles of Weil contact elements,” Bulletin of the Irish Mathematical Society 49, 2002, pp.23–41. [6] M. Kureš and D. Sehnal, “The order of algebras with nontrivial fixed point subalgebras,”Lobachevskii Journal of Mathematics 25, 2007, pp. 187–198. [7] M. Kureš, “The composition of polynomials by the substitution principle,” Journal of Discrete Mathematical Sciences & Cryptography 13, No. 6, 2010, pp. 543–552. [8] M. Kureš, “Fixed point subalgebras of Weil algebras: from geometric to algebraic questions,”in Complex and Differential Geometry, Springer Proceedings of Mathematics, 2011, pp. 183–192. 214 Cop y ri g ht © 2012 SciRes. |