Computations with relative extensions of number fields with an application to the construction of Hilbert class fields

Abstract

We present new and improved algorithms for computations with relative extensions of algebraic number fields. Especially, the tasks of relative normal forms, relative bases, detection of subfields, and embedding of these subfields are discussed. The new methods are then used to compute Hilbert class fields of totally real cubic and quartic fields for the first time.

On the construction of class fields

“…The aim of this paper is to describe new methods for computing class fields with an emphasis on the problem of tabulating extensions of number fields. While the overall strategy is the same as in [CDyDO98] and [DP95], we show how the individual steps can be improved tremendously. The theoretical improvements are accompanied by an efficient implementation allowing computations in situations which were out of reach before.…”

Kant V4

“…Accessible from KASH is a SQL-database for number fields (Daberkow and Weber, 1996). The database is designed to give easy and fast access to several hundreds of thousands of number fields.…”

On Computations in Kummer Extensions

“…Kummer extensions play a major role in the theoretical treatment of relative extensions, and especially within class field theory. By using these extensions it is, for example, possible to construct class fields of an arbitrary algebraic number field (Daberkow and Pohst, 1995;?, 1998). In a recent paper, M. Pohst described the computation of a system of relative generators for the ring of integers for a Kummer extension (Pohst, 1996).…”