Hermite and Smith normal form algorithms over Dedekind domains

We show how the usual algorithms valid over Euclidean domains, such as the Hermite Normal Form, the modular Hermite Normal Form and the Smith Normal Form can be extended to Dedekind rings. In a sequel to this paper, we will explain the use of these algorithms for computing in relative extensions of number fields. The goal of this paper is to explain how to generalize to a Dedekind domain R many of the algorithms which are usually associated with a Euclidean domain, such as the Hermite Normal Form algorithm (including a modular version), and the Smith Normal Form algorithm. Since the goal of this paper is eminently practical, we will restrict our attention to the case where R is the ring of integers of a number field, for which we assume known a Z-basis. Most of the algorithms can however be transposed to a more general context. An immediate application of these algorithms (which was evidently our sole motivation) is to computing in relative extensions of number fields. This can now indeed be done very easily, as we will show in a subsequent paper ([3]). These ideas have already been used by Bosma and Pohst [1]. Notations: R will always denote the ring of integers ZK of a number field K (although most of the results apply to general Dedekind domains), and K is the field of fractions of R. Unless otherwise specified, an ideal of R will always mean a nonzero fractional ideal.