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Fundamental Results and Algorithms in Dedekind Domains
- Source :
- Advanced Topics in Computional Number Theory ISBN: 9780387987279
- Publication Year :
- 2000
- Publisher :
- Springer New York, 2000.
-
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).
Details
- ISBN :
- 978-0-387-98727-9
- ISBNs :
- 9780387987279
- Database :
- OpenAIRE
- Journal :
- Advanced Topics in Computional Number Theory ISBN: 9780387987279
- Accession number :
- edsair.doi...........e0a7179cf4cb84156d92be24fc91742c