The software package KANT V4 for computations in algebraic number fields is now available in version 4. In addition a new user interface has been released. We will outline the features of this new software package.

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.

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Abstract. Let F be an algebraic number field and E a quadratic extension with E = F( √ µ). We describe a minimal set of elements for generating the integral elements o E of E as an o F module. A consequence of this theoretical result is an algorithm for constructing such a set. The construction yields a simple procedure for computing an integral basis of E as well. In the last section, we present examples of relative integral bases which were computed with the new algorithm and also give some running times. PreliminariesThe computation of integral bases of algebraic number fields is one of the basic tasks in computational algebraic number theory. Nonetheless, the existing algorithms for this problem tend to be very slow for fields of higher degree. In this paper, we therefore outline a new algorithm for the computation of an integral basis for those fields E which contain a subfield F of index 2.Quadratic extensions have been extensively studied in the past [8,6] and the main result of this paper is a generalization of a result of Sommer [10], who investigated biquadratic extensions.In the sequel we consider number fields F with [F : Q] = n and E subject towith an integral nonsquare element µ of F. It is well known that the ring of integers o E of E is not a free o F module, in general. The following theorem gives a necessary and sufficient criterion for the existence of a relative integral basis [1].

Let k be an algebraic number field containing a primitive mth root of unity. An extension K = k( m √ µ) of k with µ ∈ k is called a Kummer extension. These extensions have been studied extensively in the past and they play an important role in class field theory. Recently many new algorithms dealing with Kummer extensions emerged. In this paper we will give algorithms to solve two problems, which are of particular interest; the computation of the relative discriminant d K/k and the computation of Hilbert norm symbols.