F. y Diaz

Computing ray class groups, conductors and discriminants

We use the algorithmic computation of exact sequences of Abelian groups to compute the complete structure of (Z K /m) * for an ideal m of a number field K, as well as ray class groups of number fields, and conductors and discriminants of the corresponding Abelian extensions. As an application we give several number fields with discriminants less than previously known ones.

A table of quintic number fields

All algebraic number fields F of degree 5 and absolute discriminant less than 2 x 107 (totally real fields), respectively 5 x 106 (other signatures) are determined. We describe the methods which we applied and list significant data.

Subexponential algorithms for class group and unit computations

Abstract We describe in detail the implementation of an algorithm which computes the class group and the unit group of a general number field, and solves the principal ideal problem. The basic ideas of this algorithm are due to J. Buchmann. New ideas are the use of LLL-reduction of an ideal in a given direction which replaces the notion of neighbour, and the use of complex logarithmic embeddings of elements which plays a crucial role. Heuristically the algorithm performs in sub-exponential time with respect to the discriminant for fixed degree, and performs well in practice.

The minimum discriminant of totally real octic fields

Abstract The minimum discriminant of totally real octic algebraic number fields is determined. It is 282,300,416 and belongs to the ray class field over Q (√2) of conductor (7 + 2 √2): F = Q (√α) for α = (7 + 2 √2 + (1 + √2) √7 + 2 √2)/2. There is—up to isomorphy—only one field of that discriminant. The next two smallest discriminant values are 309,593,125 and 324,000,000. For each field we present a full system of fundamental units and its class number.

Tables of octic fields with a quartic subfield

We describe the computation of extended tables of degree 8 fields with a quartic subfield, using class field theory. In particular we find the minimum discriminants for all signatures and for all the possible Galois groups. We also discuss some phenomena and statistics discovered while making the tables, such as the occurrence of 11 non-isomorphic number fields having the same discriminant, or several pairs of non-isomorphic number fields having the same Dedekind zeta function.

Imprimitive ninth-degree number fields with small discriminants

We present tables of ninth-degree nonprimitive (i.e., containing a cubic subfield) number fields. Each table corresponds to one signature, except for fields with signature (3,3), for which we give two different tables depending on the signature of the cubic subfield. Details related to the computation of the tables are given, as well as information about the CPU time used, the number of polynomials that we deal with, etc. For each field in the tables, we give its discriminant, the discriminant of its cubic subfields, the relative polynomial generating the field over one of its cubic subfields, the corresponding (irreducible) polynomial over Q, and the Galois group of the Galois closure. Fields having interesting properties are studied in more detail, especially those associated with sextic number fields having a class group divisible by 3

Quadratic fields with 3-rank equal to 4

In [21 there is reference to 119 known imaginary quadratic fields that have 3-rank r > 4. We examine these fields and determine the exact values of r. Their associated real filelds and the distribution of their 3-Sylow subgroups are also studied. Some of the class groups are recorded since they are of special interest. These include examples having an infinite class field tower and only one ramified prime, and others having an infinite tower because of two different components of

A table of totally real quintic number fields

We give a table of the 1077 totally real number fields of degree five having a discriminant less than 2 000 000. There are two nonisomorphic fields of discriminant 1 810 969 and two nonisomorphic fields of discriminant 1 891 377. All the other number fields in the table are characterized by their discriminant. Among these fields, three are cyclic and four have a Galois closure whose Galois group is the dihedral group D5 . The Galois closure for all the other fields in the table has a Galois group isomorphic to S5 .

Supplement to A Table of Totally Real Quintic Number Fields

The first five columns of the table give, in this order, the coefficients a1, a2, a3, a4, a s of the polynomial P (X) = X5+a 1X4+a2X3+a3X2+a4X+a . The next column gives the discriminant dK of the field K, and is followed by the index f of the ring Z [0] in ZK: one thus has dp= f2dK, where dp is the discriminant of P (X). Finally, the decomposition in prime factors of the discriminant of the field is given, where the exponents are placed betwcen parentheses if they are greater than 1.