Let k be an algebraic number field. We describe a procedure for computing the Hilbert class field 1(k) of k, i.e., the maximal abelian extension unramified at all places. In the first part of the paper we outline the underlying theory and in the second part we present the important algorithms and give several examples.
1998 Academic Press
1. PRELIMINARIES
The computation of Hilbert class fields has been an important issue in algebraic number theory in this century. After the connection between the j-function and the Hilbert class field for imaginary quadratic fields was established, various authors [ShiTa, Sta, Deu] improved and extended this analytic way of constructing (Hilbert) class fields. However, it is not yet clear whether this approach is suitable for arbitrary ground fields.
Following the precedence of Hasse [Ha], we describe a way of computing the Hilbert class field of an algebraic number field along the lines of the proof of the existence theorem of class field theory by Kummer extensions. This is a purely algebraic approach to the problem and can be used unconditionally. The algorithm outlined below has been implemented under the computer algebra system KASH [Kant].
In the sequel we consider an algebraic number field k and compute the maximal abelian extension 1(k) of k which is unramified at all places. This field 1(k) is called the Hilbert class field of k. Before we develop an algorithm for the computation of 1(k), we need to introduce some notation and state some theorems of the class field theory which are of importance for this work. We adopt the notation of [Janu]. We note, that in the special case of Hilbert class fields the conductor is always 1. A subgroup H of the group I k of all fractional ideals of k always consists of the ideals of I k which are contained in the ideal classes of a subgroup H of the class group Article No. NT972208