Let be an algebraic number field given by the minimal polynomial of . We want to determine all subfields of given degree. It is convenient to describe each subfield by a pair such that is the minimal polynomial of . There is a bijection between the block systems of the Galois group of and the subfields of . 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 -adic methods. We give a detailed description for all parts of the algorithm.
Soit un corps de nombres défini par le polynôme minimal de . Nous nous intéressons à déterminer les sous-corps de degré donné. Chaque sous-corps est décrit en donnant le polynôme minimal de et le plongement de dans donné par un polynôme tel que . Il y a une bijection entre les systèmes de blocs du groupe de Galois de et les sous-corps de . 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 -adiques. Nous présentons ici une description détaillée de l’algorithme.
@article{JTNB_1998__10_2_243_0, 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}, publisher = {Universit\'e Bordeaux I}, volume = {10}, number = {2}, year = {1998}, doi = {10.5802/jtnb.227}, zbl = {0935.11047}, mrnumber = {1828244}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.227/} }
TY - JOUR TI - On computing subfields. A detailed description of the algorithm JO - Journal de Théorie des Nombres de Bordeaux PY - 1998 DA - 1998/// SP - 243 EP - 271 VL - 10 IS - 2 PB - Université Bordeaux I UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.227/ UR - https://zbmath.org/?q=an%3A0935.11047 UR - https://www.ams.org/mathscinet-getitem?mr=1828244 UR - https://doi.org/10.5802/jtnb.227 DO - 10.5802/jtnb.227 LA - en ID - JTNB_1998__10_2_243_0 ER -
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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