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Computing ray class groups, conductors and discriminants. (English) Zbl 0929.11064

The authors develop methods for computing invariants of number fields related to class field computations. They use ideas of their earlier paper on finitely generated Abelian groups [“Algorithmic Methods for Finitely Generated Abelian Groups”, submitted to Math. Comput.] being represented by a set of generators and a matrix of relations. Linear algebra – essentially calculations of Smith and Hermite normal forms – yield methods for computing exact sequences of such groups. These are applied to the computation of the multiplicative structure of the residue class ring \(\mathbb{Z}_K/ {\mathcal M}\) of the ring of integers of an algebraic number field \(K\) and a modulus \({\mathcal M}\) of \(K\). To solve this task it suffices to compute \((1+ {\mathfrak p}^a)/(1+ {\mathfrak p}^b)\) for prime ideals \({\mathfrak p}\) dividing \({\mathcal M}\) and finitely many suitable pairs of exponents \(a,b\) subject to \(1\leq a\leq b\leq 2a\) up to \(b=v_{\mathfrak p} ({\mathcal M})\). Consequently the task of computing ray class groups is also solved with these ideas.
In section 3 the authors then develop methods for calculating congruence groups, discriminants, signatures and conductors of ray class fields. Finally, they show how to determine defining equations for these fields. The last section contains two explicit numerical examples with class fields of total degree 12, one over a quadratic and the other over a cubic base field. A nice and interesting application is made to the computation of number fields with small discriminants and degree less than 100.

MSC:

11Y40 Algebraic number theory computations
11R37 Class field theory
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