A survey of computational class field theory
Journal de Théorie des Nombres de Bordeaux, Volume 11 (1999) no. 1, pp. 1-13.

We give a survey of computational class field theory. We first explain how to compute ray class groups and discriminants of the corresponding ray class fields. We then explain the three main methods in use for computing an equation for the class fields themselves: Kummer theory, Stark units and complex multiplication. Using these techniques we can construct many new number fields, including fields of very small root discriminant.

Le but de cet article est de décrire les avancées récentes dans la théorie algorithmique du corps de classes. Nous expliquons comment calculer les groupes de classes de rayon ainsi que les discriminants des corps de classes correspondants. Nous donnons ensuite les trois méthodes principales utilisées pour le calcul des équations des corps de classes : la théorie de Kummer, les unités de Stark et la multiplication complexe. En utilisant ces techniques, nous avons pu construire de nombreux nouveaux corps de nombres intéressants, en particulier ayant un discriminant très proche des bornes d'odlyzko.

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Henri Cohen. A survey of computational class field theory. Journal de Théorie des Nombres de Bordeaux, Volume 11 (1999) no. 1, pp. 1-13. doi : 10.5802/jtnb.235. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.235/

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