Fundamental Results and Algorithms in Dedekind Domains

Abstract

The easiest way to start studying number fields is to consider them per se, as absolute extensions of √; this is, for example, what we have done in [Coh0]. In practice, however, number fields are frequently not given in this way. One of the most common other ways is to give a number field as a relative extension, in other words as an algebra L/K over some base field K that is not necessarily equal to √. necessarily equal to ℚ. In this case, the basic algebraic objects such as the ring of integers ℤ _L and the ideals of ℤ _L , are not only ℤ-modules, but also ℤ _K- modules. The ℤ _K -module structure is much richer and must be preserved. No matter what means are chosen to compute ℤ _L , we have the problem of representing the result. Indeed, here we have a basic stumbling block: considered as ℤ-modules, ℤ _L or ideals of ℤ _L are free and hence may be represented by ℤ-bases, for instance using the Hermite normal form (HNF); see, for example, [Coh0, Chapter 2]. This theory can easily be generalized by replacing ℤ with any other explicitly computable Euclidean domain and, under certain additional conditions, to a principal ideal domain (PID). In general, ℤ _K is not a PID, however, and hence there is no reason for ℤ _L to be a free module over ℤ _K- A simple example is given by K = ℚ(√-10) and L = K(√1) (see Exercise 22 of Chapter 2).