Showing papers on "Field (mathematics) published in 1980"

Base change for gl(2)

Abstract: R. Langlands shows, in analogy with Artin's original treatment of one-dimensional p, that at least for tetrahedral p, L(s, p) is equal to the L-function L(s, ?) attached to a cuspdidal automorphic representation of the group GL(2, /A), /A being the adele ring of the field, and L(s, ?), whose definition is ultimately due to Hecke, is known to be entire. The main result, from which the existence of ? follows, is that it is always possible to transfer automorphic representations of GL(2) over one number field to representations over a cyclic extension of the field. The tools he employs here are the trace formula and harmonic analysis on the group GL(2) over a local field.

On the Stickelberger Ideal and the Circular Units of an Abelian Field.

Abstract: c* is a rational number which we study in some detail. The definition of c k not involve the class number h*. In w 3 we show that S annihilates the ideal class group of k. Finally, we remark that if k is real, the above construction trivializes; we have k= k +, A = S = s(G) 7I, and h* -c *= 1. Let E denote the units of k. In w we define a subgroup C of E, which we call the circular units of k. We show that C has finite index in E, and that this index may be written in the form [E: C]=h +.c~, where h + is the class number of k +, and c~ is a rational number whose definition does not involve h+. '* and c~. As in the In the remainder of this paper, we study the numbers c k

Minimal Models of Rational Surfaces Over Arbitrary Fields

Abstract: In this article all the types of minimal models of smooth rational surfaces defined over an arbitrary field are described. Bibliography: 19 titles.

P-adic Analysis: A Short Course on Recent Work

Abstract: This introduction to recent work in p-adic analysis and number theory will make accessible to a relatively general audience the efforts of a number of mathematicians over the last five years. After reviewing the basics (the construction of p-adic numbers and the p-adic analog of the complex number field, power series and Newton polygons), the author develops the properties of p-adic Dirichlet L-series using p-adic measures and integration. p-adic gamma functions are introduced, and their relationship to L-series is explored. Analogies with the corresponding complex analytic case are stressed. Then a formula for Gauss sums in terms of the p-adic gamma function is proved using the cohomology of Fermat and Artin-Schreier curves. Graduate students and research workers in number theory, algebraic geometry and parts of algebra and analysis will welcome this account of current research.

Solution Fields of Nonlinear Equations and Continuation Methods

Abstract: Contihuation methods are considered here in the broad sense as algorithms for the computational analysis of specified parts of the solution field of equations of the form $Fx = b$ , where $F:R^m \to R^n $ is a given mapping and $m > n$. Such problems arise, for instance, in structural mechanics and then usually $m - n$ of the variables $x_i $ are designated as parameters. For the case $m = n + 1$ an existence theory for the regular curves of the solution field is developed here. Then approximate solutions are considered and shown to be solutions of certain perturbed problems. These results are used to prove that for the continuation methods with Eider-predictor and Newton-corrector a particular steplength algorithm is guaranteed to trace any regular solution of the field. Some numerical aspects of the procedure are discussed and a numerical example is included to illustrate the effectiveness of the approach.

Implementation of Multiplication, Modulo a Prime Number, with Applications to Number Theoretic Transforms

Abstract: This paper discusses a technique for multiplying numbers, modulo a prime number, using look-up tables stored in read-only memories. The application is in the computation of number theoretic transforms implemented in a ring which is isomorphic to a direct sum of several Galois fields, parallel computations being performed in each field.

Multiples of Points on Elliptic Curves and Continued Fractions

Abstract: -6ax*-8bx + c), (1.1)with respec tto an ar^adic valuation an,d the geometry of the ellipticcurve(1.2)where we assume that a,b,c belon to ag field k. The curve (1.2) has tworational points 0 and P at infinity. Takin 0g as the origin of the Mordell-Weil grou op f (1.2) w, e find that the multiples of P in thi s grou arpeintimately connected with the convergents and complete quotients of thecontinued fraction o y.fAlthough our arguments are purely algebraic w,e are able to recovertheorems of Abel [1] and Chebychev [3], prove byd analysing the periodsof integrals, to the effect that the continued fraction of y is periodi if ancdonly if the point P is of finite order. Ther ise a very simple relationshipbetween the period of y and the order o P.f The results we prove are validover any field and can be extende tdo curves defined ove ar ring (such asthe rational integers Z) I. n this way they can be applied to the reductionmod various primes of an elliptic curve defined ove Z orr other rings.We now describe our notation. Let C be any curve define kd ove withra non-singular ^-rationa 0. Lel point = K k(C)t be the correspondingfunction field and denote the valuation correspondin to 0 b ordgy

The witt ring of a space of orderings

Abstract: The theory of "space of orderings" generalizes the reduced theory of quadratic forms over fields (or, more generally, over semilocal rings). The category of spaces of orderings is equivalent to a certain category of "abstract Witt rings". In the particular case of the space of orderings of a formally real field K, the corresponding abstract Witt ring is just the reduced Witt ring of K. In this paper it is proved that if X = (X, G) is any space of orderings with Witt ring W(X), and g: X —* Z is any continuous function, then g is represented by an element of W(X) if and only if 2oSK g(o) = 0 mod | V\ holds for all finite fans V Q X. This gener- alizes a recent field theoretic result of Becker and Brocker. Following the proof of this, applications are given to the computation of the stability index of X, and to the representation of continuous functions g: X -* ±1 by elements of G. The major result in this paper is Theorem 5.5 (the representation theorem). Theorems 6.4 and 7.2 and Corollary 7.5 are applications of this theorem. The first three sections of the paper are of an introductory nature. Most of the material in these sections is either implicit in (M2) or (M3) or is contained in the unpublished (Ml). The application of the abstract theory presented here to the Witt ring of a field is explained in some detail in Theorems 1.3 and 2.6. Theorems 3.6 and 4.5 are included to establish that Theorem 5.5 does, in fact, yield (BB, 5.3) as a special case. For the more complicated application of this theory to the Witt ring of a semilocal ring, the reader is referred to (KR, 6.7, 2.24, 2.30).

The modular curves X0(65) and X0(91) and rational isogeny

Abstract: Let N be an integer ≥ 1. The affine modular curve Y0(N) parameterizes isomorphism classes of pairs (E; F), where E is an elliptic curve defined over ℂ, the field of complex numbers, and F is a cyclic subgroup of order N. The compacti-fication X0(N) is an algebraic curve defined over ℚ.

Hamburger-Noether Expansions and Approximate Roots of Polynomials.

Abstract: Let k be a field. To any pair (x,y) ≠ (0,0) of elements in tk[[t]], and hence to any irreducible and residually rational power series f ∈ k[[X,Y]], we associate a matrix, the Hamburger-Noether tableau of (x,y), which, in essence, is a description of the algorithm used to compute an element of minimal order in the integral closure of k[[x,y]] by successive quadratic transformations. Our main aim then is to reprove basic results of Abhyankar and Moh on approximate roots of polynomials and their use in the study of branches of plane curves, and to extend these results to certain situations over fields of positive characteristic where one important technical tool of Abhyankar and Moh, the Newton-Puiseux expansion of a branch, is not available.

Annales de la faculté des sciences detoulouse

Abstract: Summary : A linear independence measure is given for the coordinates of algebraic points of abe-lian functions in two variables. From this an abelian analogue of the Franklin-Schneider theoremis deduced.Let A be a simple abelian variety defined over the field of algebraic numbers and let0 : ~2 -~A be a normalised theta homomorphism (cf. [12], § 1.2). be entire func-tions such that (~~(z),...,~v(z)) forms a system of homogeneous coordinates for the point 0(z) in projective v-space. Put fi : = ~i/~~. Assume that ~~(o) ~ 0 ; then algebraic for all i. A point u in C =~ 0 is by definition an algebraic point of 0 if and only if f.(u) is algebraic forall i. The field of abelian functions associated with 0 is ~ (fi ,...,f~). If (ul,u2) is a non-zero algebraic point of 0, the coordinates ul and u2 are linearlyindependent over the algebraic numbers (cf. [12], Theoreme 3.2.1) ; the proof uses the Schneider- Lang criterion (cf. [5] , Chapter II I,

SL(2,5) and Frobenius Galois Groups Over Q

Abstract: A finite transitive permutation group G is called a Frobenius group if every element of G other than 1 leaves at most one letter fixed, and some element of G other than 1 leaves some letter fixed. It is proved in [20] (and sketched below) that if k is a number field such that SL(2, 5) and one other nonsolvable group Ŝ5 of order 240 are realizable as Galois groups over k, then every Frobenius group is realizable over k. It was also proved in [20] that there exists a quadratic (imaginary) field over which these two groups are realizable. In this paper we prove that they are realizable over the rationals Q, hence we Obtain THEOREM 1. Every Frobenius group is realizable as the Galois group of an extension of the rational numbers Q.

Recursive algorithms for the matrix Padé problem

Abstract: . A matrix triangularization interpretation is given for the recursive algorithms computing the Padd approximants along a certain path in the Padd table, which makes it possible to unify all known algorithms in this field [51, [61. For the normal Padd table, all these results carry over to the matrix Padd problem in a straightforward way. Additional features, resulting from the noncommutativity are investigated. A generalization of the Trench-Zohar algorithm and related recursions are studied in greater detail.

The Galois group of a polynomial with two indeterminate coefficients

Abstract: Φ 0) is a polynomial in which two of the coefficients are indeterminates t, u and the remainder belong to a field F. We find the galois group of / over F(t, u). In particular, it is the full symmetric group Sn provided that (as is obviously necessary) /(X) Φ fχ(Xr) for any r > 1. The results are always valid if F has characteristic zero and hold under mild conditions involving the characteristic of F otherwise. Work of Uchida [10] and Smith [9] is extended even in the case of trinomials Xn + tXa + u on which they concentrated. 1* Introduction* Let F be any field and suppose that it has characteristic p, where p — 0 or is a prime. In [9], J. H. Smith, extending work of K. Uchida [10], proved that, if n and a are coprime positive integers with n > α, then the trinomial Xn + tXa + u, where t and u are independent indeterminates, has galois group Sn over F(t, u), a proviso being that, if p > 0, then p \ na(n — α). (Note, however, that this conveys no information whenever p — 2, for example.) Smith also conjectured that, subject to appropriate restriction involving the characteristic, the following holds. Let I be a subset (including 0) of the set {0, 1, , n — 1} having cardinality at least 2 and such that the members of / together with n are co-prime. Let T — {ti9 i e 1} be a set of indeterminate s. Then the polynomial Xn + ΣΓ^o1*^* has galois group Sn over F(T). In this paper, we shall confirm this conjecture under mild conditions involving p(>0), thereby extending even the range of validity of the trinomial theorem. In fact, we also relax the other assumptions. Specifically, we allow some of the tt to be fixed nonzero members of F and insist only that two members of T be indeterminates. Indeed, even if the co-prime condition is dispensed with, so that the galois group is definitely not SΛ, we can still describe

The determination of all imaginary, quartic, abelian number fields with class number 1

Abstract: In this paper, it is proved that there are just seven imaginary number fields, quartic cyclic over the rational field, and having class number 1. These are the quartic, cyclic imaginary subfields of the cyclotomic fields generated by the fth roots of unity, where f is 16 or is a prime less than 100. This completes the list of imaginary, quartic, abelian number fields with class number 1. There are 54 such fields, with maximal conductor 67.163. In [5] , Uchida proves the following about imaginary, quartic abelian number fields of class number 1: (A) If the field is bicyclic, there are 47 such fields, (with maximal conductor 67.163) with possibly one more. (B) If the field is cyclic, the conductor must be less than 50000. The existence of another field in (A), depends, however, on there being an imaginary quadratic field with class number 2 and discriminant <-427. Stark [4], and Montgomery-Weinberger [3], have shown there is no such quadratic field, so Uchida's list of bicyclic fields is complete. Brown and Parry [1] , using the results of Stark, have also given a complete list of imaginary bicyclic quartic fields of class number 1. This paper describes the computations that were carried out to show that there are just 7 imaginary, quartic cyclic number fields with class number 1. THEOREM. There are exactly 54 imaginary, quartic, abelian number fields with class number 1: (a) 47 bicyclic fields, with maximal conductor 67.163; and (b) 7 cyclic fields, with maximal conductor 61. As will be seen below, the cyclic fields are contained in the cyclotomic fields 16 Q5 Q13' Q29 Q37, Q53, and Q61 (QN is the field generated by the Nth roots of unity.) Each is the unique imaginary, cyclic, quartic subfield of the corresponding cyclotomic field. The theory of the relative class number developed by Hasse [2], was used to eliminate all possible fields up to the 50000 limit, except for the above seven. We describe the details relevant to the present problem. Let k be an imaginary, cyclic quartic field. There is, associated to k, a pair of quartic Dirichlet characters a, and i4i of conductor f = conductor of k. These are, essentially, the characters of order 4 on Gal(QfIQ) o (Z/f)*, which are orthogonal to Gal(Qf/k). Since k is totally complex, Received September 6, 1979. 1980 Mathematics Subject Classification. Primary 12A30. ? 1980 American Mathematical Society 002 5-57 18/80/0000-0180/$02.0

Abelian extensions of regular local rings

Abstract: The integral closure of a regular local ring in a finite Abelian extension of its quotient field is Cohen-Macaulay, provided that the degree of the extension is not divisible by the characteristic of the residue field. THEOREM. Let R be a regular local ring, K its quotient field, and L a finite Galois extension of K with Abelian Galois group, which we denote G. Assume that the order of G is not divisible by the characteristic of the residue field of R. Then if S is the integral closure of R in L, S is Cohen-Macaulay. PROOF. We first note that since L is assumed to be a Galois extension of K, and thus separable, S is a finitely generated R-module. Since S is integrally closed, it is therefore reflexive as an R-module; that is, the natural map from S to S**= HomR(HomR(S, R), R) is an isomorphism (see Bourbaki [1, Chapter 7, ?4.8]). The action of the Galois group G on L makes L into a K[G]-module; since any K-automorphism of L must preserve the integral elements over R, S similarly becomes an R[G]-module. The usual isomorphism: S ?R K -L can then be considered as an isomorphism of K[ G ]-modules. The Normal Basis Theorem then says that there is an element of L whose conjugates form a basis for L over K; or, in other words, that L is isomorphic to K[G] as a K[G]-module. Hence S is an R[G]-module such that S ? R K K[G] as K[G]-module. We wish to show that S is a Cohen-Macaulay ring, and we will do this by showing that it is isomorphic to R' (where n is the degree of L over K) as an R-module. We first reduce to the case where R is complete with algebraically closed residue field. LEMMA. Let R -> R' be a faithfully flat extension of regular local rings. Then if S= S ?RR' and K' is the quotient field of R', we have: (a) S' is a reflexive R'-module. (b) If S' = (R')' as R'-module, then SR' as R-module. (c) S' is an R'[G]-module such that S' ?R' K'K'[G] as K'[G]-module. PROOF OF LEMMA. Since R' is flat over R, the natural map from HomR(M, N) ?R R' to HomR,(R' ?R M, R' ?R N) is an isomorphism whenever M is a finitely presented R-module; hence, applying this twice, S** OR R'-(S')** is an isomorphism. Thus the diagram Received by the editors July 12, 1978 and, in revised form, November 30,1978. AMS (MOS) subject classifications (1970). Primary 13H10; Secondary 13B05. ? 1980 American Mathematical Society 0002-9939/80/0000-0102/$02.00

Intersections of real closed fields

Abstract: 1. In this paper we wish to study fields which can be written as inter­ sections of real closed fields. Several more restrictive classes of fields have received careful study (real closed fields by Artin and Schreier, hered­ itarily euclidean fields by Prestel and Ziegler [8], hereditarily Pythago­ rean fields by Becker [1]), with this more general class of fields sometimes mentioned in passing. We shall give several characterizations of this class in the next two sections. In § 2 we will be concerned with Gal(F/F), the Galois group of an algebraic closure F over F. We also relate the fields to the existence of multiplier sequences; these are infinite sequences of elements from the field which have nice properties with respect to certain sets of polynomials. For the real numbers, they are related to entire functions; generalizations can be found in [3]. In § 3 a characteriza­ tion is given in terms of finite Galois extensions of the field. This is applied in § 4 to show that these fields suffice to obtain all isomorphism classes of reduced Witt rings (of equivalence classes of anisotropic quadratic forms over a field) with a certain finiteness condition on the rings. In this section we shall briefly outline some of the work other authors have done with these and related classes of fields. Our interest is only in formally real fields, though to study them we shall often have to look at their algebraic extensions. For any formally real field F, we denote by F* the intersection of all the real closed subfields of a fixed algebraic closure F which contain F. These fields have been studied in [6] where they are called "galois order closed" because of the following theorem.

Zur Chevalley-Zerlegung von Derivationen

Abstract: Let k be a valued field of characteristic zero. We consider analytic k-algebras, i.e. finite algebras over rings of convergent power series over k, and their k-derivations. The following theorem is proved: Let k be algebraically closed and A a normal analytic k-algebra of dimension 2. If there is a k-derivation on A not acting nilpotently, then A is homogeneous, i.e. a residue class ring of a power series ring by a (weighted) homogeneous ideal.

A dimension theorem for division rings

Abstract: LetD be a division algebra over a fieldk, letn be an arbitrary positive integer, and letk(x1,...,xn) denote the rational function field inn variables overk. In this note we complete previous work by proving that the following three conditions are equivalent: (i) there exists an integerj such that the matrix ringMj(D) contains a commutative subfield which has transcendence degreen overk; (ii) K dim (D⊗kk(x1,...,xn)) =n; (iii) gl. dim (D⊗kk(x1,...,xn)) =n. The crucial tool in the proof of this theorem is the Nullstellensatz forD[x1,...,xn] which was obtained by Amitsur and Small.

Tannaka duality for discrete groups

Abstract: Introduction. Let A be a commutative ring, G a group, and let RePA(G) be the category of representations pv:G AutA(V) of G on finitely generated modules over A. We shall deal with the question to what extent G can be recovered from RepA(G). Note that Pontryagin's classical duality theorem asserts, essentially, that every compact abelian group can be reconstructed from its irreducible unitary representations with the operation of tensor product between them (namely, the group of the characters). Following Pontryagin, several duality theories (i.e., theorems about the possibility and a way to reconstruct a group from its representations) were established. The first one was the "Duality of Tannaka" for compact Lie groups [24] (see also [71). The work of Tannaka was generalized in several directions (see for example [1, 6, 10, 171). The principal work for Lie groups was done by Hochschild and Mostow [11]. They used the algebra of representative functions RF(G) (the F-linear span of the coefficients of all the finite dimensional representations of G over a field F) and the group MF(G) of proper automorphisms of RF(G) (i.e., automorphisms which fix the constant functions and commute with the G-right translations of RF(G)). From their results in [11] one may deduce the following Theorem which is cited here in order to illustrate what we mean by Tannaka duality:

The graph-theoretic field model— II. application of multi-terminal representations to field problems

Abstract: This paper is a sequel to a paper entitled “The Graph-Theoretic Field Model—I: Modelling and Formulations” (1). Herein, the Theory of Multi-Terminal Representations is applied to the Graph-Theoretic Field Model to provide mathematical models of finite elements. The element models are obtained solely from the algebraic building blocks of the Graph-Theoretic Field Model, without recourse to any functional mathematics. The theory of Multi-Terminal Representations is developed for both linear and non-linear problems. Examples of the application of the theory to one- and two-dimensional field problems are presented from heat conduction and electrostatics.

Field-strength description of non-Abelian gauge theories

Abstract: The method of dual transformation developed by Sugamoto is applied to the SU(2) pure Yang-Mills theory and the SO(3) Georgi-Glashow model After the dual transformation, the partition functions are expressed only in terms of field strengths

Geometric field theories with a given set of constant solutions

Abstract: Given a system of matter and geometric fields defined on the ten-dimensional spacej of the local reference frames, we find the conditions which have to be satisfied by the Lagrangian if we require that the theory has a given set of constant solutions. The constant solutions describe empty regions, where the spacej can be considered as the manifold of a Lie group. We consider with more details the case in which all the constant solutions are generated by the action of a symmetry group on a given constant solution. We analyse a large class of symmetry groups and we find, as particular cases, some Lagrangians which have been proposed and studied recently. We find also some interesting theories symmetric with respect to a de SitterSO3,2 orSO4,1 group. They contain a microscopic fundamental length and have a nonlocal character.