Showing papers on "Field (mathematics) published in 1984"

Equivariant stable homotopy and Segal's Burnside ring conjecture

Abstract: Analogous results were proved later for KO, in the generality of compact Lie groups, by Atiyah and Segal [8], and for KFq, the algebraic K-theory spectrum associated to the finite field Fq, by Rector [29] using Quillen's [26] computation of iT*(KFq). In each case, the answer involves an appropriate completed representation ring of G, and the cohomology theory in question is constructed from the permutative category of finite dimensional vector spaces over a field (see [31]). If one considers cohomology theories constructed from other permutative categories, one expects to find analogous computations in terms of a "completed representation ring" of G in the given category, appropriately defined. In particular, stable cohomotopy, Ts*, is constructed from the category of finite sets [31], and for this category the analogue of the representation ring is a well-known object, the Burnside ring A(G) [13]. A(G) is a commutative ring with augmentation, so one may speak of A(G), the completed Burnside ring. Moreover, there is

On L -functions of elliptic curves and cyclotomic towers

Abstract: In this paper we shall strengthen a recent theorem of Ralph Greenberg [1] concerning L-functions of elliptic curves and anticyclotomic towers. Let K be an imaginary quadratic field and M an abelian extension of K, possibly of infinite degree. We say that the extension M/K is anticyclotomic if M is Galois over Q and if the nontrivial element of Gal(K/Q) acts on Gal(M/K) by inversion. Now fix a finite set P of rational primes and consider the compositum (in an algebraic closure of K) of all anticyclotomic extensions of K which are unramified outside P. We denote this compositum by L and refer to it as the maximal anticyclotomic extension of K unramified outside P. The field L may also be described as the union of all ring class fields of K with conductor divisible only by primes in P. Class field theory shows that Gal(L/K) is isomorphic to the product of a finite group and the group

On integrability properties of SU (2) Yang–Mills fields. I. Infinitesimal part

Abstract: We study the distribution of 2‐plane elements on which infinitesimal parallel displacement of isovectors yields identity. This yields to an algebraic and differential classification and, in the generic case, to a quasimetric naturally associated with the field.

Remarks on Tarski's problem concerning (R, +, *, exp)

Abstract: Publisher Summary The chapter presents the elementary theory of the structure (R , + , .), and the results could be extended to the structure (R, +, ., exp). Some aspects of on (R , + , .) are reviewed and its usage is inquired. The decidability of Th(R , + , .) is a nice result in its own right and quite useful in many theoretical decidability questions but has otherwise not been important in settling open problems. Th(R , + , .·)= theory of real closed fields is useful in proving properties of real closed fields: in certain cases the only known proof consists of first establishing the property for the field of reals by transcendental methods and then invoking elimination of quantifiers for (R ,

Parallel algorithms for algebraic problems

Abstract: Fast parallel algorithms are presented for the following problems in symbolic manipulation of univariate polynomials: computing all entries of the extended Euclidean scheme of two polynomials over an arbitrary field, gcd and 1cm of many polynomials, factoring polynomials over finite fields, and the squarefree decomposition of polynomials over fields of characteristic zero and over finite fields.For the following estimates, assume that the input polynomials have degree at most n, and the finite field has $p^d $ elements. The Euclidean algorithm is deterministic and runs in parallel time $O(\log ^2 n)$. All the other algorithms are probabilistic (Las Vegas) in the general case, but when applicable to ${\bf Q}$ or ${\bf R}$, they can be implemented deterministically over these fields. The algorithms for gcd and lcm use parallel time $O(\log ^2 n)$. The factoring algorithm runs in parallel time $O(\log ^2 n\log ^2 (d + 1)\log p)$. The algorithm for squarefree decomposition runs in parallel time $O(\log ^2 n)$...

The 4-class ranks of quadratic fields

Abstract: Let K be a quadratic extension field of the rational numbers Q. Let C K be the 2-class group of K in the narrow sense. It is a classical result that rank C K= t l , where t is the number of primes that ramify in K/Q. Now let C~ = {ai: a e CK}, and let R K denote the 4-class rank of K in the narrow sense; i.e., RK= rank ~Kc z = dimv2 ttc'Z/t~41~K/~KJ" Here F 2 is the finite field with two elements, and 2 4 CK/C K is an elementary abelian 2-group which we are viewing as a vector space o v e r F 2. Given a quadratic field K, one can compute R K by computing the rank (over F2) of a certain matrix of Legendre symbols (cf. [11]). Now assume K is imaginary quadratic. So K=Q(1/~mm), where m is a square-free positive integer. For each positive integer t, each nonnegative integer e, and each positive real number x, we define

Computing Logarithms in Finite Fields of Characteristic Two

Abstract: A simple algorithm to find logarithms in a finite field of characteristic two is described. It uses the Euclidean algorithm for polynomials in attempting to reduce an element to a product of factors all of whose logarithms are stored in a database. The algorithm, which is similar to one of Adleman, has a random runtime and constant storage requirements. It is analyzed and problems associated with the construction of the database are considered. The aim of the work is to show that the algorithm is feasible for the field with $2^{127} $ elements on which several proposed public key distribution systems have been based. For such application it is felt that the discrete logarithm is still a viable technique for sufficiently large fields.

Free Subgroups of Units in Group Rings

Abstract: In this paper we give necessary and sufficient conditions under which the group of units of a group ring of a finite group G over a field K does not contain a free subgroup of rank 2. Some extensions of this results to infinite nilpotent and FC groups are also considered.

On places of algebraic function fields.

Abstract: In this paper we study the space of all places of a function field in n variables. We denote by F/k an (algebraic) function field, i.e. F is a fmitely generated extension of k of transcendence degree n ̂ l . By a place P of F/k we mean a place of F which is trivial on k, i. e. the restriction of P to k yields an isomorphism. The image of χ e F under P will be denoted by xP; correspondingly, FP denotes the set of all images of elements of F. Thus FP consists of the residue field of the valuation vp associated to P, together with oo, the value which is assigned to χ by P, if χ has a \"pole at P\". Nevertheless, in the following we will simply call FP the residue field of P. The degree of transcendency of FP over k is called the dimension of P. It is denoted by dim(P). In defining dim(P) we already have identified k with its isomorphic image under P. This will be done frequently in the following sections. The ordered abelian group of values taken by vp will be denoted by vP(F). The rational rank of P is the dimension of vP(F)

Commutative Algebraic Groups and Intersections of Quadrics

Abstract: Let E be a commutative connected algebraic group over a field k. The purpose of this paper is to construct embeddings of E into some pN such that the homogeneous ideal of the closure E of E in ~,u is generated by quadratic forms. In particular such an E is idealtheoretically an intersection of quadrics. To outline a special case let k be the field of complex numbers and consider the canonical exact sequence O-~G ~ E-~ A ~O

On the fundamental units and the class numbers of real quadratic fields

Abstract: Let Q be the rational number field and h(m) be the class number of the real quadratic field with a positive square-free integer m. It is known that if h(m) = 1 holds, then m is one of the following four types with prime numbers p ≡ 1, pt ≡ 3 (mod 4) (1 昤 i ≥ 4) : i) m = p, ii) m = p1 , iii) m = 2 or m = 2p2 , iv) m = p3p4 (see Behrbohm and Redei [1]). The sufficient conditions for h(m) > 1 with these m were given by several authors: in all cases by Hasse [2], in case i) by Ankeny, Chowla and Hasse [3] and by Lang [4], in case ii) by Takeuchi [5] and by Yokoi [6].

Remarks on the orbital analytic classification of germs of vector fields

Abstract: Associated to the germ of a holomorphic vector field on whose linear part belongs to a Siegel domain, is the germ of a conformal map ; the latter is the monodromy transformation induced by a circuit around the singular point on a separatrix.It is proved that the monodromy transformations are moduli for the orbital analytic classification of germs of vector fields at a singular point: two vector field germs with the same linear part of Siegel type are orbitally analytically equivalent if and only if for each of the germs one can choose a local separatrix such that these separatrices are tangent at zero and such that the monodromy maps corresponding to them are analytically equivalent.Moduli for the orbital analytic classification of vector field germs in higher-dimensional spaces are also constructed, and a new proof of the theorem about the topological classification of vector fields with saddle resonant singular points is given.Bibliography: 24 titles.

On the Complexity of the Groebner-Bases Algorithm over K[x, y, z]

Abstract: In /Bu65/, /Bu70/, /Bu76/ B. Buchberger presented an algorithm which, given a basis for an ideal in K[x1,...,xn] (the ring of polynomials in n indeterminates over the field K), constructs a so-called Grobner-basis for the ideal. The importance of Grobner-bases for effectively carrying out a large number of construction and decision problems in polynomial ideal theory has been investigated in /Bu65/, /Wi78/, /WB81/, /Bu83b/. For the case of two variables B. Buchberger /Bu79/, /Bu83a/ gave bounds for the degrees of the polynomials which are generated by the Grobner-bases algorithm. However, no bound has been known until now for the case of more than two variables. In this paper we give such a bound for the case of three variables.

Group Rings and Division Rings

Abstract: Continuing the work in [ll],[l2] we study division algebras D = k(G) over a field k which are generated by some polycyclic-by-finite subgroup G of the multiplicative group D* of D. We discuss a specific class of examples of such division algebras that can be thought of as multiplicative analogs of the Weyl field. Furthermore, we show that the division algebras D = k(G) always contain free subalgebras of rank ≤ 2, provided G is not abelian-by-finite. Finally, we discuss some open questions concerning commutative subfields and Lie commutators in D = k(G).

Conjugacy classes of hyperbolic matrices in (,) and ideal classes in an order

Abstract: A bijection is proved between Sl(7s,Z?-conjugacy classes of hyperbolic matrices with eigenvalues { A1, . . ., An} which are units in an n-degree number field, and narrow ideal classes of the ring R;, = Z[A,]. A bijection between Gl(n, Z)-conjugacy classes and the wide ideal classes, which had been known, is repeated with a different proof. In 1980, Peter Sarnak was able to obtain an estimate of the growth of the class number of real quadratic number fields using the Selberg Trace Formula and a bijection between hyperbolic elements of S1(2, Z) and quadratic forms. The "class number" counted was the number of congruence classes of quadratic forms as studied by Gauss [1]. In this paper we will translate this bijection into modern number-theoretic terms by counting ideal classes in a ring of integers associated to a given field. In this way a bijection is proved between conjugacy classes of hyperbolic matrices in S1(2, Z) with a given set of eigenvalues and ideal classes in a certain order (i.e. subring of dimension n over Z) associated to the ring of integers OK in a real n th degree number field K. This more direct method is necessary for generalizing the bijection to higher dimensional cases because Sarnak's result depends upon quadratic forms, Pell's equation and other things which are well understood only ill the case of S1(2, Z). We must mention the work of Latimer and MacDuffee [3] who first proved Theorem 2 in a slightly different fashion. Important also is the extensive work of Taussky [7-9], who simplified the results of Latimer and MacDuffee and extended them in certain directions, as well as doing much work on the S1(2, Z) case. It follows from a brief examination of the characteristic polynomial for a matrix A in Sl(n, Z) that the eigenvalues of A are conjugate units in an extension of Q. We shall insist in the remainder of this paper that A be "hyperbolic" with irreducible characteristic polynomial, that is, A will have distinct real eigenvalues A('), each of which is of degree n over Q. PROPOSITION 1. If A is an eigenvalue for a matrix A E SL(n, Z), then for any field K containing A there exists an eigenvector w TS = (@1S.. ., Xn), with wi E OK. Received by the editors March 30, 1983. 1980 Mathematics Subject Classification. Primary 10-02, 15-02; Secondary 15A18, 15A36, lOC07. tw1984 American Mathematical Society 0025-5726/84 $1.00 + $.25 per page

Some theorems on the generalized numerical ranges

Abstract: Let F be the real field R the complex field C or the skew field H of real quaternions. For any c = i:c1,…,cn )∊ Rn: and for any two hermitian (symmetric) matrices A and B with elements in F, the authors give a unified proof that the set is convex (with n> 2 if F =R). They also show that the convexity problem, definite-ness problem, and inclusion problem associated with the above set, are all equivalent, and give some applications of these results.

Generalized Bernoulli numbers and $m$-regular primes

Abstract: A prime p is defined to be m-regular if p does not divide the class number of a certain abelian number field. Several different characterizations are given for a prime to be m-regular, including a description in terms of the generalized Bernoulli numbers. A summary is given of two computations which determine the m-regularity or m-irregularity of primes p for certain values of m and p. In an earlier article [3], we defined m-regular primes and showed under certain simple congruence conditions that the Fermat equation of exponent p has no solutions in the field Q(vm ) when p is an m-regular prime. In this note we give several equivalent conditions for a prime to be m-regular. In particular, it is possible to describe m-regularity by means of the generalized Bernoulli numbers. The following notation will be used throughout this article. Q: the field of rational numbers. Z: ring of rational integers. m: a square-free integer, m # 1. p: an odd prime. g: a primitive pth root of unity. k = Q(xm): quadratic number field. L =Q(; + t K= Q(;, 'm ) K1: the maximal real subfield of K. K3 = Q('): the pth cyclotomic field. K2: the totally imaginary quadratic extension of L contained in K with K2 + K3. h ): the class number of the field ( ). ( ): the character group of the field (. X: a character in K. f(x): the conductor of X. d = ImrI or 41rm1: the conductor of k. The prime p is defined to be m-regular if p does not divide h (K). Since m may be replaced with thte squart-fre keuDei c4 1)(P pm, we may assume that (m, p)= 1. Received January 17, 1983; revised August 4, 1983 and October 11, 1983. 1980 Mathematics Subject Classification. Primary 12A50, 12A35, 12A15. ?P 1984 American Mathematical Society 0025-5718/84 $1.00 + $.25 per page

Cubic subfields of exceptional simple Jordan algebras

Abstract: Let E/k be a cubic field extension and J a simple exceptional Jordan algebra of degree 3 over k. Then E is a reducing field of J if and only if E is isomorphic to a (maximal) subfield of some isotope of J. If k has characteristic not 2 or 3 and contains the third roots of unity then every simple exceptional Jordan division algebra of degree 3 over k contains a cyclic cubic subfield.

Using Fields for Semiring Computations

Abstract: We construct a field within which computations for the semi-ring (Z∪{-∞}, max, +) can be carried out using classical algebra, and apply it to studying a range of problems over this semiring

A q-analogous of the characterization of hypercubes as graphs

Abstract: We give an arithmetical characterization of graphs which are realizable as graphs of the lattice ℒ of the subspaces of a graphic (or projective) space of finite dimension and of finite order q ≥ 1. In other words ℒ is any complemented modular lattice of finite rank and of finite order q. When q ≥ 2 and the rank is at least four, ℒ is the lattice of the subspaces of a finite dimensional vector space over the field GF(q). Two independent axioms are required involving the number of geodesies between any two vertices. This number must be a simple function of the distance.