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Computational class field theory. (English) Zbl 1177.11095
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, 497-534 (2008).
The authors give a survey on computational methods in class field theory. In the introduction the classical Theorem of Kronecker and Weber is presented as motivation. Section 2 contains a short overview on the results of class field theory, especially Artin’s reciprocity law. This is followed by an introduction into ideles and their importance with respect to class fields. Sections 4 and 5 cover the computational aspects of class field theory using Kummer extensions. The remaining sections treat the computation of class fields over imaginary quadratic fields via complex multiplication. The authors give a modern treatment of Weber’s ideas for generating such fields by class invariants using Shimura’s reciprocity law. This yields generating equations with much smaller coefficients than by values of the $j$-function.
##### MSC:
 11Y40 Algebraic number theory computations 11R37 Class field theory for global fields
##### Keywords:
computation of class fields