Michael Pohst

Computational algebraic number theory

Intorduction.- Topics from finite field.- Topics from the geometry of number.- Algebraic number field.- Computation of an integral basis.- Computation of the unit group.- Computation of the class group.- 1 The number field sieve.- 2 KANT.- References.

A modification of the LLL reduction algorithm

The reduction algorithm of Lenstra et al. (1982) is modified in a way that the input vectors can be linearly dependent. The output consists of a basis of the lattice generated by the input vectors as well as non-trivial linear combinations of O by the input vectors if those are linearly dependent.

On the computation of number fields of small discriminants including the minimum discriminants of sixth degree fields

Abstract A new method of determining algebraic number fields with discriminants of small absolute value is developed that avoids lengthy considerations of subfields. As an application all minimum discriminants of sixth degree fields are computed.

On effective computation of fundamental units. II

Introduction. In part I of this paper we described the theoretical improvements of our method of computing fundamental units in algebraic number fields which we achieved in the last few years. Those improvements were of considerable influence on the corresponding computer program. Since its first 1976 [2] implementation it was completely rewritten and changed in so many details that a new presentation cannot be avoided. In Sections 1 and 2 of this paper we describe the applications of part I [6] to the algorithm for constructing fundamental units. Section 3 contains a complete list of numerical examples concerning algebraic number fields of small degree and small absolute discriminant. Besides the fundamental units the tables contain much information about fields of degree five and six which was so far unknown. The determination of those fields (and their subfields) is described in [3]. Besides the fundamental units we also listed the order of the torsion subgroup TUF of the unit group U, whenever it is different from 2. It was computed by the methods of Section 2 of [5]. All computations were carried out on the Control Data Cyber 76 of the Computer Center of the University of Cologne.

Constructing integral lattices with prescribed minimum. II

Methods for computing integral laminated lattices with prescribed minimum are developed. Laminating is a process of stacking layers of an (n 1)-dimensional lattice as densely as possible to obtain an n-dimensional lattice. Our side conditions are: All scalar products of lattice vectors are rational integers, and all lattices are generated by vectors of prescribed minimum (square) length et. For m = 3 all such lattices are determined.

On the resolution of index form equations in biquadratic number fields, II

In this paper we develop a method for computing all small solutions (i.e. with coordinates of absolute value <107) of index form equations in totally real biquadratic number fields. If the index form equation is not solvable, this will also be recognized by our algorithm in most cases. As an application we present all such solutions in quadratic extensions K of Q(√5) of discriminant DKQ < 63000 and of Q(√2) of discriminant DKQ < 39000.

On maximal finite irreducible subgroups of (,). I. The five and seven dimensional cases

General methods for the determination of maximal finite absolutely irreducible subgroups of GL(n, Z) are described. For n = 5, 7 all these groups are computed up to Z-equivalence.

Computing the multiplicative group of residue class rings

Let k be a global field with maximal order 0k and let m0 be an ideal of 0k. We present algorithms for the computation of the multiplicative group (0k/m0)* of the residue class ring 0k/m0 and the discrete logarithm therein based on the explicit representation of the group of principal units. We show how these algorithms can be combined with other methods in order to obtain more efficient algorithms. They are applied to the computation of the ray class group Clkm modulo m = m0m∞, where m∞ denotes a formal product of real infinite places, and also to the computation of conductors of ideal class groups and of discriminants and genera of class fields.

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.

Regulatorabschätzungen für total reelle algebraische Zahlkörper

Abstract A lower bound for the regulator of a totally real algebraic number field is determined. The regulator occurs in the determinant of a suitable positive definite quadratic form, and the desired bound is obtained by estimating the minimum of this quadratic form from below. This can be done by solving an extremal value problem with subsidiary conditions from the properties of the units of the field. Numerous examples (Tables I, III) illustrate the advantage of this method over known results.