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The arithmetic of number rings. (English) Zbl 1216.11099

Buhler, J. P. (ed.) et al., Algorithmic number theory. Lattices, number fields, curves and cryptography. Cambridge: Cambridge University Press (ISBN 978-0-521-80854-5/hbk). Mathematical Sciences Research Institute Publications 44, 209-266 (2008).
Let \(K\) be a number field. Almost all the information that number theorists are interested in are defined within the ring \({\mathcal O}_K\) of integers in \(K\). From the algorithmic point of view this has the drawback that all this information is difficult to get because finding \({\mathcal O}_K\) requires the factorization of the square part of the discriminant disc \(K\), which can turn out to be quite difficult if the discriminant is large. There are other reasons for considering not necessarily maximal orders: their relation to maximal orders is analogous to that of singular curves to their nonsingular models; in addition, orders of the form \(\mathbb Z[\alpha]\) occur in many investigations, for example in the number field sieve.
Despite these reasons for studying general orders there is no discussion of nonmaximal orders in the literature aimed at algebraic number theorists, and this article fills this gap. The following keywords should give an impression of the scope of this text: localization, regular, primary and invertible ideals, normalization and integral closure, linear algebra over \(\mathbb Z\) and resultants, geometry of numbers, zeta functions, completions of number fields, the adelic and idelic point of view, Arakelov class groups, decomposition and inertia subgroups, and Chebotarev’s density theorem.
Despite its depth this article can (and should) be read by anyone familiar with some basic commutative algebra covered by undergraduate courses in algebraic geometry and a solid background in algebraic number theory.
For the entire collection see [Zbl 1154.11002].

MSC:

11R04 Algebraic numbers; rings of algebraic integers
11Y40 Algebraic number theory computations
11Y16 Number-theoretic algorithms; complexity
11R29 Class numbers, class groups, discriminants
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