Displaying similar documents to “The non-normal quartic CM-fields and the dihedral octic CM-fields with ideal class groups of exponent ≤ 2 ”

On 2 -class field towers of imaginary quadratic number fields

Franz Lemmermeyer (1994)

Journal de théorie des nombres de Bordeaux

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For a number field k , let k 1 denote its Hilbert 2 -class field, and put k 2 = ( k 1 ) 1 . We will determine all imaginary quadratic number fields k such that G = G a l ( k 2 / k ) is abelian or metacyclic, and we will give G in terms of generators and relations.

Maximal unramified extensions of imaginary quadratic number fields of small conductors, II

Ken Yamamura (2001)

Journal de théorie des nombres de Bordeaux

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In the previous paper [15], we determined the structure of the Galois groups Gal ( K u r / K ) of the maximal unramified extensions K u r of imaginary quadratic number fields K of conductors ≦ 1000 under the Generalized Riemann Hypothesis (GRH) except for 23 fields (these are of conductors ≧ 723 ) and give a table of Gal ( K u r / K ) . We update the table (under GRH). For 19 exceptional fields K of them, we determine Gal ( K u r / K ) . In particular, for K = 𝐐 ( - 856 ) , we obtain Gal ( K u r / K ) ≅ S 4 ˜ × C 5 and K u r = K 4 , the fourth Hilbert class field of K . This is the first example of a number...

A simple characterization of principal ideal domains

Clifford S. Queen (1993)

Acta Arithmetica

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1. Introduction. In this note we give necessary and sufficient conditions for an integral domain to be a principal ideal domain. Curiously, these conditions are similar to those that characterize Euclidean domains. In Section 2 we establish notation, discuss related results and prove our theorem. Finally, in Section 3 we give two nontrivial applications to real quadratic number fields.

The size function h 0 for quadratic number fields

Paolo Francini (2001)

Journal de théorie des nombres de Bordeaux

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We study the quadratic case of a conjecture made by Van der Geer and Schoof about the behaviour of certain functions which are defined over the group of Arakelov divisors of a number field. These functions correspond to the standard function h 0 for divisors of algebraic curves and we prove that they reach their maximum value for principal Arakelov divisors and nowhere else. Moreover, we consider a function k 0 ˜ , which is an analogue of exp h 0 defined on the class group, and we show it also...

Cryptography based on number fields with large regulator

Johannes Buchmann, Markus Maurer, Bodo Möller (2000)

Journal de théorie des nombres de Bordeaux

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We explain a variant of the Fiat-Shamir identification and signature protocol that is based on the intractability of computing generators of principal ideals in algebraic number fields. We also show how to use the Cohen-Lenstra-Martinet heuristics for class groups to construct number fields in which computing generators of principal ideals is intractable.

Maximal unramified extensions of imaginary quadratic number fields of small conductors

Ken Yamamura (1997)

Journal de théorie des nombres de Bordeaux

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We determine the structures of the Galois groups Gal ( K u r / K ) of the maximal unramified extensions K u r of imaginary quadratic number fields K of conductors ≦ 420 ( ≦ 719 under the Generalized Riemann Hypothesis). For all such K , K u r is K , the Hilbert class field of K , the second Hilbert class field of K , or the third Hilbert class field of K . The use of Odlyzko’s discriminant bounds and information on the structure of class groups obtained by using the action of Galois groups on class groups is essential. We...