An investigation of bounds for the regulator of quadratic fields.
Another look at real quadratic fields of relative class number 1
The relative class number of a real quadratic field K = ℚ (√m) of discriminant d is defined to be the ratio of the class numbers of and , where denotes the ring of integers of K and is the order of conductor f given by . R. Mollin has shown recently that almost all real quadratic fields have relative class number 1 for some conductor. In this paper we give a characterization of real quadratic fields with relative class number 1 through an elementary approach considering the cases when...
Arithmetic of Pell surfaces
Arithmetische Eigenschaften hyperbolischer Klassen elliptischer Modulgruppen.
Artin's primitive root conjecture for quadratic fields
Fix an element in a quadratic field . Define as the set of rational primes , for which has maximal order modulo . Under the assumption of the generalized Riemann hypothesis, we show that has a density. Moreover, we give necessary and sufficient conditions for the density of to be positive.
Atkin-Lehner involutions and class number residuality
Bases normales d'entiers et unités modulaires.
Bemerkungen zum Satz von Heegner-Stark über die imaginär-quadratischen Zahlkörper mit der Klassenzahl Eins.
Berechnung von Einheiten nach Lagrange.
Bernoulli numbers, Hurwitz numbers, p-adic L-functions and Kummer's criterion.
Let K = Q(ζp) and let hp be its class number. Kummer showed that p divides hp if and only if p divides the numerator of some Bernoulli number. In this expository note we discuss the generalizations of this type of criterion to totally real fields and quadratic imaginary fields.
Bestimmung einklassiger Geschlechter ternärer quadratischer Formen in reell-quadratischen Zahlkörpern.
Bicyclic commutator quotients with one non-elementary component
For any number field with non-elementary -class group , , the punctured capitulation type of in its unramified cyclic cubic extensions , , is an orbit under the action of . By means of Artin’s reciprocity law, the arithmetical invariant is translated to the punctured transfer kernel type of the automorphism group of the second Hilbert -class field of . A classification of finite -groups with low order and bicyclic commutator quotient , , according to the algebraic invariant...
Binary forms and unramified An-extensions of quadratic fields.
Calcul du nombre de classes d'un corps quadratique imaginaire ou réel, d'après Shanks, Williams, McCurley, A. K. Lenstra et Schnorr
Dans cette note nous décrivons différentes méthodes utilisées en pratique pour calculer le nombre de classes d'un corps quadratique imaginaire ou réel ainsi que pour calculer le régulateur d'un corps quadratique réel. En particulier nous décrivons l'infrastructure de Shanks ainsi que la méthode sous-exponentielle de McCurley.
Chowla's conjecture
Class fields towers of imaginary quadratic fields.
Class groups of large ranks in biquadratic fields
For any integer , we provide a parametric family of biquadratic fields with class groups having -rank at least 2. Moreover, in some cases, the -rank is bigger than 4.
Class number and factorization in quadratic number fields
Class number formulae for imaginary quadratic fields.
Class number formulae for imaginary quadratic fields. II.