On computing subfields. A detailed description of the algorithm
Journal de Théorie des Nombres de Bordeaux, Volume 10 (1998) no. 2, pp. 243-271.

Let (α) be an algebraic number field given by the minimal polynomial f of α. We want to determine all subfields (β)(α) of given degree. It is convenient to describe each subfield by a pair (g,h)[t]×[t] such that g is the minimal polynomial of β=h(α). There is a bijection between the block systems of the Galois group of f and the subfields of (α). These block systems are computed using cyclic subgroups of the Galois group which we get from the Dedekind criterion. When a block system is known we compute the corresponding subfield using p-adic methods. We give a detailed description for all parts of the algorithm.

Soit (α) un corps de nombres défini par le polynôme minimal de α. Nous nous intéressons à déterminer les sous-corps (β)(α) de degré donné. Chaque sous-corps est décrit en donnant le polynôme minimal g de β et le plongement de β dans (α) donné par un polynôme h tel que h(α)=β. Il y a une bijection entre les systèmes de blocs du groupe de Galois de f et les sous-corps de (α). Ces systèmes de blocs sont calculés en utilisant les sous-groupes cycliques du groupe de Galois qui sont obtenus à partir du critère de Dedekind. Lorsqu’un système de blocs est connu, on calcule le sous-corps correspondants par des méthodes p-adiques. Nous présentons ici une description détaillée de l’algorithme.

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Jürgen Klüners. On computing subfields. A detailed description of the algorithm. Journal de Théorie des Nombres de Bordeaux, Volume 10 (1998) no. 2, pp. 243-271. doi : 10.5802/jtnb.227. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.227/

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