Displaying similar documents to “Maximal unramified extensions of imaginary quadratic number fields of small conductors, II”

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...

Unramified quaternion extensions of quadratic number fields

Franz Lemmermeyer (1997)

Journal de théorie des nombres de Bordeaux

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Classical results of Rédei, Reichardt and Scholz show that unramified cyclic quartic extensions of quadratic number fields k correspond to certain factorizations of its discriminant disc k . In this paper we extend their results to unramified quaternion extensions of k which are normal over ℚ , and show how to construct them explicitly.

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...