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).KeywordsPrime IdealNumber FieldProjective ModuleIntegral IdealTorsion ModuleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.