Francisco y Diaz

A Survey of Discriminant Counting

We give a survey of known results on the asymptotic and exact enumeration of discriminants of number fields, both in the absolute and relative case. We give no proofs, and refer instead to the bibliography.

A Table of Totally Complex Number Fields of Small Discriminants

Using the explicit class field theory developed in [3] and tables of number fields in low degree, we construct totally complex number fields having a degree smaller than 80 and a root discriminant near from Odlyzkos bounds. For some degrees, we extend and improve the table of totally complex number fields of small discriminants given by Martinet

Counting Discriminants of Number Fields of Degree up to Four

For each permutation group G on n letters with n ≤ 4, we give results, conjectures and numerical computations on discriminants of number fields L of degree n over ℚ such that the Galois group of the Galois closure of L is isomorphic to G.

Computing Ray Class Groups, Conductors and Discriminants

We describe the computation of ray class groups of number fields, conductors and discriminants of the corresponding Abelian extensions. As an application we give several number fields with discriminants less than previously known ones.

Constructing complete tables of quartic fields using Kummer theory

We explain how to construct tables of quartic fields of discriminant less than or equal to a given bound in an efficient manner using Kummer theory, instead of the traditional (and much less efficient) method using the geometry of numbers. As an application, we describe the computation of quartic fields of discriminant up to 107, the corresponding table being available by anonymous ftp.

Cyclotomic extensions of number fields

Abstract Let K be a number field, l a prime number, ζ l a primitive l -th root of unity and Kz = K(ζ l ). In this paper, we first give a detailed description of the discriminant, conductor, different and prime ideal decomposition of the extension Kz/K. We apply this to obtain the Galois-module structure of certain finite modules associated to prime ideals above l , and we also give the Galois-module structure of the unit group of Kz modulo l th powers.

Signed fundamental domains for totally real number fields

We give a signed fundamental domain for the action on

Algorithmic Methods for Finitely Generated Abelian Groups

\mathbb{R}^n_+

Construction of Tables of Quartic Number Fields

of the totally positive units

Computation of Relative Quadratic Class Groups

E_+