Showing papers in "Crelle's Journal in 1967"

On Artin's conjecture.

Abstract: The problem of determining the prime numbers p for which a given number a is a primitive root, modulo JP, is mentioned, for the partieular case a — 10, by Gauss in the section of the Disquisitiones Arithmeticae that is devoted to the periodie deeimal expansions of fraetions with denominator p. Several writers in the nineteenth Century subsequently alluded to the problem, but, since their results were for the most part of an inconclusive nature, we are content to single out from their work the interesting theorem that 2 is a primitive root, modulo p, if p be of the form 4g + l, where q is a prime. The early work, however, was eonfined almost entirely to special cases, it not being until 1927 that the problem was formulated definitely in a general sense. In the latter year the late Emil Artin enunciated the celebrated hypothesis, now usually known äs Artin's conjecture, that for any given non-zero integer a other than l, —l, or a perfect square there exist infinitely many primes p for which is a primitive root, modulo p. Furthermore, letting Na(#) denote the number of such primes up to the limit #, he was led to conjecture an asymptotic formula of the form

On Epstein's Zeta-function.

Abstract: This paper was written in the Spring of 1949, and a resume appeared in the note: On Epstein's zeta Function (I), Proceedings of the National Acadeiny of Sciences (U. S. A.), 35 (1949), 371--374. Meanwhile, the following papers which have reference to the Proceedings paper, came to our attention: 1. J.B. Rosser, Real roots of real Dirichlet L-series, Jour. Research National Bureau of Standards, 46 (1950), 505—514. 2. E. A. Anferteva, On an identity of Chowla and Seiberg (Russian), Izvestija Vyssik Ucebnyh Zavadenii Mathematika (Kazan), No. 3 (10) (1959), 13—21. 3. P. T. Bateman and E. Grosswald, On Epstein's zeta Function, Acta Arithmetica, 9 (1964), 365—373. 4. K. Ramachandra, Some applications of Kronecker's limit formulas, Annals of Mathematics 80 (1964), 104—148.

Minimally 2-connected graphs.

Abstract: Definition 1. A graph is called 2-connected if it contains at least 2 vertices and each pair of vertices belong to some circuit contained in the graph. For graphs with at least 3 vertices this is equivalent to each of the following [2] (1) The graph is connected and contains no cut-vertex. (2) The graph is connected and each pair of edges belong to some circuit contained in the graph. (3) The graph consists of one member. The equivalence of these four properties will be assumed freely throughout this paper.

On the varieties of special divisors on a curve.

Abstract: In the classical literature one repeatedly finds assertions to the effect that the set of special linear series of degree n and dimension r on a curve of genus g, if not empty, \"depends on (r + 1) (n — r) — r g parameters\". The result is stated already in Riemann's \". . . AbeFschen Functionen\". The primary purpose of the present paper is to give a more precise formulation of this fact in terms of the dimension of a certain algebraic subset of the jacobian variety of the curve. The Statement and proof of the result will be seen to be independent of the groundfield, and the result is therefore valid in arbitrary characteristic. More speeifically, let X be a complete non-singular curve of genus g Ξ> 2, let J(X) be its jacobian variety, and let φ ι X-+ J(X) be a canonical map which we normalize once and for all by selecting a point P € X and setting φ (P) = 0 (the identity element of /(X)). If D = ΣrriiQi is a positive divisor on X, we define r, we can now formulate our main result.

On the euler and bernoulli polynomials

Abstract: [2], [3] that E5(x) = Ix — -^ (x — χ — l), so at least one example is known of an \ Ι Euler polynomial with a multiple factor. Also, it has been proven by Carlitz [2] that Ep(x)l ix — -^\ is irreducible for p a prime == 3 (mod 4), and that E2p(x)lx(x — 1) has an irreducible factor of degree at least p — l for p a prime. In the case of the Bernoulli polynomials, Inkeri [8] has shown that the only Bn(x), n ^> 3, possessing rational roots are B2m+l(x) for m ̂ l, which have the well-known factors χ (x — -^-) (x — 1). Other than these, there have until now been no non-trivial factors of any degree known for any Bn(x). l 1\ On the other hand, Carlitz [2] has shown that B2m+l(x)jx [x — -^ (x — 1) for \ A/ 2m + l — k(p — 1) + l? A fS p, p an odd prime, must possess an irreducible factor of degree at least 2m + l — P· In the same paper it is also proven that B2m(x) is irreducible for 2m = k (p — 1) ,̂ p a prime, t ̂ 0 and l <Ξ k < p. More recently McCarthy [10] has shown that B2m(x) is irreducible for 2m = (k p + A + 1) (p — IX A < p. In our investigation we will first develop various properties of the Euler polynomials. This will then be followed by a parallel development for the Bernoulli polynomials, which is made possible in large part by a rather surprising modular relationship between the two sets of polynomials. In particular, for the Euler polynomials we will determine their rational and multiple roots, s well s the roots that lie on the lines x = 0, -^ , and l in the complex plane. We will also obtain partial formulas for their discriminants, the manner in which

Abelian curves of small conductor.

Abstract: The equation is minimal at a prime p if each coefficient a^ is integral at j9, but ordp(Zl) is äs small äs possible. Since the ring of integers Z of Q is a principal ideal domain, it is easy to see that we can find & global minimal Weierstrass model for A over Q, i. e. a cubic equation äs above in which each aj € Z , simultaneously minimal at all p. At each p, we have a well-defined reduced curve A by reducing the coefficients (of a minimal model) modulo p. If A is non-singular, i. e. if ( ) — 0, then A is an abelian curve in charaeteristic p and we say A has non-degenerate reduction at p ; if A is singular, then the set of non-singular points on A is in a natural way a multiplicative or additive group (defined over the algebraic closure of the field of p elements, let us say), and we say A has multiplicative or additive reduction, respectively. Regarding A äs being defined over a suitably large finite unramified extension of the jp-adic field Qp, we also have a reduction in the sense of Neron (cf. [4]) of A at p and we denote by np the number of components, not counting multiplicities. It is proved in [5] that the quantity ordp(A) + 1 — np is 0 for non-degenerate reduction, l for multiplicative reduction, and at least 2 for additive

Eine Idealstruktur Banachscher Operatoralgebren.

Abstract: J. W. Calkin [3] beantwortete die Frage nach den zweiseitigen, abgeschlossenen Idealen in der Banachalgebra ä(H) der beschränkten linearen Transformationen eines separablen Hilbertraumes, indem er die vollstetigen Operatoren als das einzige derartige Ideal in £(H) charakterisierte. Der Operatorenring £,(H) ist für allgemeine, nicht notwendig separable Hilberträume von Interesse, da er nach dem Einbettungssatz von L M. Gelfand und M. A. Neumark [6] die universelle C*-Algebra darstellt: Jede C*Algebra, eine symmetrische Banachalgebra mit der Eigenschaft \\\\xx* || = ||#||, ist isometrisch isomorph einer in der Operatornorm abgeschlossenen Teilalgebra von £ (H) bei geeignet gewähltem Hilbertraum H. Über C^-Algebren gibt es beginnend mit den Arbeiten von J. v. Neumann [13] über schwach abgeschlossene C*-Algebren eine umfassende Literatur. Wesentliche Beiträge zur Idealtheorie von C*-Algebren lieferten I. E. Segal [19], L Kaplanski [9] und R. T. Prosser [15]. Die Idealstruktur bzgl. zweiseitiger, abgeschlossener Ideale der C^-Algebren ä(H) bei beliebigem Hilbertraum H war bisher nicht bekannt; sie wird in der vorliegenden Arbeit angegeben: Für jede Algebra £(H) gibt es eine eindeutig bestimmte, alle zweiseitigen, abgeschlossenen Ideale enthaltende Kompositionsreihe mit minimalem und maximalem Element (Theorem 3. 3). Die dazu entwickelte Methode beruht auf einer formalen Verallgemeinerung der Kompaktheit, indem eine Mächtigkeitsbedingung hinzugefügt wird (Definition 1. 2, 1. 5). Eine andere, in gewissem Sinne komplementäre Verallgemeinerung findet man in der Literatur (z. B. Köthe [11], S. 20) als ^-Kompaktheit, wobei #„ eine Schranke für die Mächtigkeit der zugelassenen Überdeckungssysteme darstellt, aus denen ein endliches Teilsystem ausgewählt werden kann. Im Gegensatz dazu bezieht sich der hier eingeführte Begriff der Kompaktheit vom Grad J$T auf die Mächtigkeit des auswählbaren Teilsystems. Daraus ist der für diese Arbeit bzw. für [7] zentrale Begriff der Präkompaktheit vom Grad XT (Definition 1. 2) in metrischen Räumen bzw. uniformen Räumen abgeleitet. Dies führt zu einer Kette zweiseitiger, abgeschlossener Ideale für Banachräume (Satz 2. l, 2. 2); aus dieser Kette gewinnt man bei Spezialisierung auf Hilberträume mit Hilfe des Spektralsatzes die erwähnte Kompositionsreihe. In Kapitel l wird die Präkompaktheit vom Grad # für metrische Räume eingeführt und in Verbindung mit der Separabilität und der Kompaktheit vom Grad # untersucht. Der letzte Begriff erscheint als eine zunächst plausiblere Verallgemeinerung der Kompaktheit, ist aber schwieriger zu handhaben; die Präkompaktheit ist für unsere Unter-

Sur les extensions cubiques non-Galoisiennes des rationnels et leur clôture Galoisienne.

Abstract: Le premier de ces buts a ete atteint gräce ä Fintroduction des ideaux «essentiels» dont l'etude fait Fobjet du premier chapitre. Dans la seconde partie on montre comment la decomposition d'un ideal essentiel en facteurs «canoniques» jointe ä l'etude prealable de la decomposition de (3) dans K et N permet de determiner explicitement une base de 0K. Dans le dernier chapitre, on utilise encore la decomposition des ideaux essentiels pour caracteriser l'existence de bases normales du Ofc-module 0N. D'une part on montre Finteret de Fetape intermediaire que constitue l'existence de bases quasi-normales de 0N: ces bases sont obtenues en adjoignant ä l deux des conjugues relativement ä k d'un entier de N. D'autre part, on met en evidence le role important joue par la trace relativement a k de 0N. Cette partie s'acheve par quelques remarques sur le Z-module 0N, remarques qui ne permettent cependant pas de caracteriser les cas oü 0N est un Z[P]module libre, ( designe le groupe de Galois de N par rapport ä Q).