On computing subfields. A detailed description of the algorithm

Jürgen Klüners

doi:10.5802/jtnb.227

Jürgen Klüners

Journal de théorie des nombres de Bordeaux, Tome 10 (1998) no. 2, pp. 243-271

Abstract (VO)
Résumé

Let $ℚ (α)$ be an algebraic number field given by the minimal polynomial $f$ of $α$ . We want to determine all subfields $ℚ (β) \subset ℚ (α)$ of given degree. It is convenient to describe each subfield by a pair $(g, h) \in ℤ [t] \times ℚ [t]$ 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 $ℚ (α)$ . 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 $ℚ (α)$ un corps de nombres défini par le polynôme minimal de $α$ . Nous nous intéressons à déterminer les sous-corps $ℚ (β) \subset ℚ (α)$ de degré donné. Chaque sous-corps est décrit en donnant le polynôme minimal $g$ de $β$ et le plongement de $β$ dans $ℚ (α)$ 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 $ℚ (α)$ . 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.

MR Zbl

DOI : 10.5802/jtnb.227

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     author = {J\"urgen Kl\"uners},
     title = {On computing subfields. {A} detailed description of the algorithm},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {243--271},
     year = {1998},
     publisher = {Universit\'e Bordeaux I},
     volume = {10},
     number = {2},
     doi = {10.5802/jtnb.227},
     zbl = {0935.11047},
     mrnumber = {1828244},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.227/}
}

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Jürgen Klüners. On computing subfields. A detailed description of the algorithm. Journal de théorie des nombres de Bordeaux, Tome 10 (1998) no. 2, pp. 243-271. doi: 10.5802/jtnb.227

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