Showing papers on "Field (mathematics) published in 1983"

Computing in Algebraic Extensions

Abstract: The aim of this chapter is an introduction to elementary algorithms in algebraic extensions, mainly over Q and, to some extent, over GF(p). We will talk about arithmetic in Q(α) and GF(p n ) in Section 1 and some polynomial algorithms with coefficients from these domains in Section 2. Then, we will consider the field K of all algebraic numbers over Q and show constructively that K indeed is a field, that multiple extensions can be replaced by single ones and that K is algebraically closed, i.e. that zeros of algebraic number polynomials will be elements of K (Section 4 — 6). For this purpose we develop a simple resultant calculus which reduces all Operations on algebraic numbers to polynomial arithmetic on long integers together with some auxiliary arithmetic on rational intervals (Section 3). Finally, we present some auxiliary algebraic number algorithms used in other chapters of this volume (Section 7). This chapter does not include any special algorithms of algebraic number theory. For an introduction and survey with an extensive bibliography the reader is referred to Zimmer [15].

On the Birch and Swinnerton-Dyer Conjecture

Abstract: Let F be a finite extension of the rational field Q. If E is an elliptic curve defined over F, then the Mordell-Weil group E(F) of points on E with coordinates in F is a finitely generated abelian group. Let L(E/F, s) be the HasseWeil zeta function of E over F which, at least in the case where E has complex multiplication, is known to be an analytic function on the entire complex plane. Birch and Swinnerton-Dyer have conjectured that L(E/F,s) has a zero at s = l of order precisely equal to the rank of E(F) over Z. The strongest result known in support of this conjecture is the following theorem of Coates and Wiles [4].

A critical-pair/completion algorithm for finitely generated ideals in rings

Abstract: In 1965, the author introduced a "critical-pair/completion" algorithm that starts from a finite set F of polynomials in K[x1,...,xn] (K a field) and produces a set G of polynomials such that the ideals generated by F and G are identical, but G is in a certain standard form (G is a "Grobner-basis"), for which a number of important decision and computability problems in polynomial ideal theory can be solved elegantly. In this paper, it is shown how the critical-pair/completion approach can be extended to general rings. One of the difficulties lies in the fact that, in general, the generators of an ideal in a ring do not naturally decompose into a "head" and a "rest" (left-hand side and right-hand side). Thus, the crucial notions of "reduction" and "critical pair" must be formulated in a new way that does not depend on any "rewrite" nature of the generators. The solution of this problem is the starting point of the paper. Furthermore, a set of reduction axioms is given, under which the correctness of the algorithm can be proven and which are preserved when passing from a ring R to the polynomial ring R[x1,...,,xn]. Z[x1,...,xn] is an important example of a ring in which the critical-pair/completion approach is possible.

Products of idempotent matrices

Abstract: A result of J. Erdos [2] states that if A is a singularn×n matrix with entries in a field F then A can be written as the product of idempotents over F. C. S. Ballantinc III quantified this result by relating the minimum number of idempotents required to the rank of A and in particular proved that A can be written as the product of n idempotents over F. In this paper we consider the case where F is replaced by a ring. We show thai if R is a division ring or a Euclidean ring, then every singular n×n matrix with entries in R can be expressed as a product of idempotents over R.

Parallel algorithms for algebraic problems

Abstract: In Borodin-von zur Gathen-Hopcroft[82] the following program is laid out: obtain a “theory package for parallel algebraic computations”, i.e. fast parallel computations for the widely used problems of symbolic manipulation in an algebraic context. In that paper, two basic problems were considered: solving systems of linear equations and computing the gcd of two polynomials, both over arbitrary ground fields. The present paper continues this program, and fast parallel solutions to the following algebraic problems are given: computing all entries of the Extended Euclidean Scheme of two polynomials over an arbitrary field, gcd and lcm of many polynomials, factoring polynomials over finite fields, and the squarefree decomposition of polynomials over fields of characteristic zero and over finite fields.

The Mordell-Weil Group of Curves of Genus 2

Abstract: In 1922 Mordell [2] proved Poirrcare’s conjecture that the group of rational points on an abelian variety of dimension 1 (= elliptic curve with rational point) is finitely generated. His proof was somewhat indirect. In 1928 Weil [5] in his thesis generalized Mordell’s result to abelian varieties of any dimension and to any algebraic number field as ground field. At the same time, Weil [6] gave a very simple and elegant proof of Mordell’s original result. I observed some time ago that Weil’s simple proof admits a further simplification. He uses the explicit form of the duplication and addition theorems on the abelian variety, but these can be avoided by, roughly speaking, the observation that every element of a cubic field-extension k(0) of a field k can be put in the shape (a0 + b)/(c0 + d) (a, b, c, d ∈ k). This additional simplification has little interest in itself, but suggested that similar ideas might be usefully exploited in investigating the Mordell-Weil group of (the Jacobians of) curves of genus greater than 1. This is done here for curves of genus 2.

On the existence of fields governing the 2-invariants of the classgroup of (√) as varies

Abstract: This paper formulates general conjectures relating the structure of the 2-classgroup C2(dp) associated to Q(fdp) to the splitting of the ideal (p) in certain algebraic number fields Here d i 2 (mod 4) is a fixed integer and p varies over primes The conjectures assert that there exists an algebraic number field Qj(d) such that the Artin symbol ((Q2(d)/Q)/(p)) determines the first j 2-invariants of the group C2(dp), ie it determines C2(dp)/C2(dp)2J These conjectures imply that the set of primes p for which C2(dp) has a given set of 2-invariants has a natural density which is a rational number Existing results prove the conjectures wheneverj = I or 2 and also for an infinite set of d withj = 3 The smallest open case isj = 3, d = -21 This paper presents evidence concerning these conjectures for d = -4, 8 and -21 Numerical evidence is given that Q23(-21) exists, and that natural densities which are rational numbers exist for the sets of primes with 2'/ h(dp) for d = -4 and 8, for I

The group of automorphisms of algebraic function fields

Abstract: There are many similar questions one can ask about both algebraic number fields and algebraic function fields. E. Freid and J. Kollar [4] and M. Fried [5] have shown that every finite group appears äs the füll automorphism group of some finite (not necessarily Galois) extension L/0. In algebraic function fields over an algebraically closed field of constants k, a complete solution to the analogue of the inverse problem is known in characteristic zero. A. Douady [3] has shown that every finite group can be realized äs a Galois group over the rational function field k(x). When k is algebraically closed of non-zero characteristic, M. Jarden [7] has shown that any given finite group appears äs a Galois group over k(x) except for finitely many characteristics.

On commutative semigroup algebras

Abstract: This paper is concerned with the problem of finding necessary and sufficient conditions on a commutative semigroup S for the algebra FS of S over a field F to be semiprimitive (Jacobson semisimple).

A new structured design method for convolutions over finite fields, Part I

Abstract: The structure of bilinear cyclic convolution algorithms is explored over finite fields. The algorithms derived are valid for any length not divisible by the field characteristic and are based upon the small length polynomial multiplication algorithms. The multiplicative complexity of these algorithms is small and depends on the field of constants. The linear transformation matrices A, B (premultiplication), and C (postmultiplication) defining the algorithm have block structures which are related to one another. The rows of A and B and the columns of C are maximal length recurrent sequences. Because of the highly regular structure of A, B , and C , the algorithms can be very easily designed even for large lengths. The application of these algorithms to the decoding of Reed-Solomon codes is also examined.

Class groups of complex quadratic fields

Abstract: We present 75 new examples of complex quadratic fields that have 5-rank of their class groups 5» 3. Only one of these fields has 5-rank of its class group > 3: The field Q(\/-25855935I511807) has a class group isomorphic to C(5) X C(5) x C(5) X C(5) X (7(2) X (7(11828). The fields were obtained by applying ideas of J. F. Mestre to the 5-isogeny ^,(11) -» Xn(\ 1).

Surveys in Combinatorics: IRREGULARITIES OF PARTITIONS: RAMSEY THEORY, UNIFORM DISTRIBUTION

Abstract: . In this survey we discuss a common feature of some classical and recent results in number theory, graph theory, etc. We try to point out the fascinating relationship between the theory of uniformly distributed sequences and Ramsey theory by formulating the main results in both fields as statements about certain irregularities of partitions. Our approach leads to some new problems as well. INTRODUCTION In 1916 Hermann Weyl published his classical paper entitled “Ober die Gleichverteilung von Zahlen mod Eins”. This was intended to furnish a deeper understanding of the results in diophantine approximation and to generalize some basic results in this field. The theory of uniformly distributed sequences has originated with this paper. In the last decades this subject has developed into an elaborate theory related to number theory, geometry, probability theory, ergodic theory, etc. Curiously enough, Issai Schur's paper entitled “Ober die Kongruenz x n +y n ≡z n (mod p)” appeared in the very same year. He proved that if the positive integers are finitely colored, then there exist x, y, z having the same color so that x+y=z. Though Ramsey theory has various germs, Schur's theorem can be regarded as the first Ramsey-type theorem. Now literally the same applies to Ramsey theory as to the theory of uniform distribution: In the last decades Ramsey theory became an elaborate theory related to number theory, geometry, probability theory, ergodic theory, etc.

When is the semigroup ring perfect

Abstract: A characterization of perfect semigroup rings A (G) is given by means of the properties of the ring A and the semigroup G. It was proved in (10) that for a ring with unity A and a group G the group ring A(G) is perfect if and only if A is perfect and G is finite. Some results on perfectness of semigroup rings were obtained by Domanov (3). He reduced the problem of describing perfect semigroup rings A(G) to checking that certain semigroup algebras derived from A(G) satisfy polynomial identities. Further, a characterization of such PI-algebras over a field of characteristic zero was found in (2). However, the obtained results are difficult to formulate and refer to some exterior constructions obscuring an insight into the properties of the semigroup. The purpose of this paper is to completely characterize perfect semigroup rings by means of the properties of the semigroup and the coefficient ring. Our approach is quite different from that of (3) and omits PI-methods. It works in arbitrary characteristic and the final result is a natural strengthening of the conditions for A(G) to be semilocal (7). In what follows A will be an associative ring, G-a semigroup. A is said to be

On the Ideal Equation I(B ∩ C) = IB∩IC

Abstract: Let R be an integral domain with quotient field K and let I be a nonzero ideal of R. We show (1) that I is invertible if and only if for every nonempty collection {Bα} of ideals of R and (2) that I is flat if and only if I(B ∩ C) = IB∩IC for each pair of ideals B and C of R.

Convolution using a conjugate symmetry property for number theoretic transforms over rings of regular integers

Abstract: In a recent paper, Dubois and Venetsanopoulos [7] have derived a method for convolving sequences over a ring S, using number theoretic transforms (NTT's) over an extension ring R of S. Until now, their assumptions have only been verified for the special case that R is an extension field of S. This paper examines their assumptions for the very general class of rings of regular integers, a class of rings that was introduced by the authors in [5]. In this paper, we also present an algorithm for the synthesis of rings suitable for number theoretic transforms and with low computational complexity.

Quotients of infinite reflection groups

Abstract: Let W be the Weyl group of a finite dimensional complex simple Lie algebra. The structure of W is quite well-known ; see [2, 3] for instance. In particular, W is finite and W/O2(W) is isomorphic to a symmetric group or an orthogonal or symplectic group over the field of two elements. It is natural to consider certain infinite analogues W(p,q,r) of such Weyl groups and inquire about their structure. These are reflection groups defined by diagrams T(p, q, r) of the form

Preorderings compatible with probability measures

Abstract: The main theorem proved in this paper is: Let B be a a-complete Boolean algebra and >- a binary relation with field B such that: (i) Every finite subalgebra B' admits a probability measure pu' such that for p, q C B', p > q iff p.'p .> 'q.

The Cartan matrix of a group algebra modulo any power of its radical

Abstract: We prove that the Cartan matrix of a group algebra F(G) modulo any power of its radical J is dual symmetric, provided F is a splitting field of F(G)/J. This eases the process of determining the Loewy series of the projective indecom- posable F(G)-modules. Let G be any finite group and F a field of characteristic p. By a module of the group algebra F(GJ, we will always mean a right module. Let S1, S2,.. ,Sk be a complete set of representatives of the isomorphism classes of simple F(G)-modules, and let P, be the projective cover of S,. For any F(G)-module M, we let M* denote the dual F(G)-module of M. The dual of S, is denoted by S,* as well. Thus P,* = Pi., as F(G) is symmetric. In fact we will only use this fundamental property of a group algebra in the following. Consequently, similar results hold for symmetric algebras rather than just group algebras. In the following we set A = F(G) and denote its radical by J. As usual, the Loewy length of an A-module M is the minimal number r for which Mjr -0. Observe that M and M* always have the same Loewy length, as M = M**. From now on we must assume F is a splitting field. Now let I be any power of J and set A = A/I, ' = P,/P,I. As S,I = 0 for all i,

Recursive FIR digital filter design using a z -transform on a finite ring

Abstract: Properties of a new complex number-theoretic z-transform (CNT z-transform) over a finite ring are presented here and related to the usual z-transform. Using the Chinese remainder theorem, it is convenient to use finite rings that are isomorphic to the direct sum of finite or Galois fields of the form GF(q2) where q is a Mersenne prime. Many properties of the usual z-transform are preserved in the CNT z-transform. This transform is used in the present paper to design both recursive and nonrecursive FIR filters on a finite ring. The advantages of the FIR filter on a finite ring are the following: 1) the absence of a roundoff error build up in the computation of either the recursive or nonrecursive realization of the filter; 2) when the FIR filter is recursive, the question of stability does not arise as long as the magnitudes of the impulse response and the input sequence do not exceed their design values; 3) for the frequency sample representation of the FIR filter an absolute error bound on the impulse response function can be obtained in terms of the power spectrum. The time required to compute a nonrecursive FIR filter on the Galois field GF(q2), where q is a Mersenne prime, is competitive with the similar nonrecursive realization on the usual complex number field, using the FFT algorithm.

Realizing algebraic number fields

Abstract: In the paper [13], the authors studied the problem of realizing rational division algebras in a special way. Let D be a division algebra that is finite dimensional over the rational field Q. If p is a prime, we say that D is p-realizable when there is a p-local torsion free abelian group A whose rank is the dimension of D over Q, such that D is isomorphic to the quasiendomorphism ring of A.

Algorithms for integrating elementary functions in terms of logarithmic integrals and error functions

Abstract: Since R Risch published an algorithm for calculating symbolic integrals of elementary functions in 1969, there has been an interest in extending his methods to include nonelementary functions In this thesis, we use the framework of differential algebra to make precise the notion of integration in terms of nonelementary or special functions Basing our work on a recent extension of Liouville's theorem on integration in finite terms, we then describe decision procedures for determining if a given element in a transcendental elementary field has an integral which can be written in terms of logarithmic integrals or error functions These algorithms first examine the structure of the integrand in order to limit the special functions which could appear in the integral to a finite number This allows us to write a general expression for the integral and then use techniques similar to those employed by Risch to calculate the undetermined parts We conclude with a demonstration of a Macsyma program for integrating a class of transcendental elementary functions in terms of elementary functions and logarithmic integrals

Brauer Factor Sets, Noether Factor Sets, and Crossed Products

Abstract: The role of Noether’s crossed products and factor sets in the study of the Brauer group Br(F) of a field F is well known. In particular, it is central in the determination of the Brauer group of a number field and in the proof of the Albert—Brauer—Hasse—Noether theorem that central division algebras over number fields are cyclic ([2], [5], [8], [9]). The central algebraic result of Noether’s theory is the isomorphism of the subgroup Br(E/F) of Br(F) consisting of the algebra classes having a finite dimensional Galois extension field E/F as a splitting field with the co-homology group H 2 (G, E*) where G = Gal E/F. This leads to an isomorphism (given later) of the full Brauer group Br(F) with a cohomology group of the Galois group of the separable algebraic closure of F.

A characterization of the semi-isometries of a Minkowski plane over a field K

Abstract: Under certain hypotheses on the field K,W.Bens has proved that a map of the plane K2, which preserves a single Lorents-Minkowski — distance is semilinear and injective [5],[6]. We shall generalise this result for every field K with char K ∉ {2,3,5 } and for K = GF(5m), m> 1.