Condition nécessaire et suffisante pour que certain groupe de Galois soit métacyclique
Annales Mathématiques Blaise Pascal, Tome 16 (2009) no. 1, pp. 83-92.

Soient d est un entier sans facteurs carrés, K=Q(d,i), i=-1, K 2 (1) le 2-corps de classes de Hilbert de K, K 2 (2) le 2-corps de classes de Hilbert de K 2 (1) et G=Gal(K 2 (2) /K) le groupe de Galois de K 2 (2) /K. Notre but est de montrer qu’il existe une forme de d tel que le 2-groupe G est non métacyclique et de donner une condition nécessaire et suffisante pour que le groupe G soit métacyclique dans le cas où d=2p avec p un nombre premier tel que p1(mod4).

Let d be positive square-free integers, K=Q(d,i) and i=-1. Let K 1 (2) be the Hilbert 2-class field of K, K 2 (2) be the Hilbert 2-class field of K 1 (2) and G=Gal(K 2 (2) /K) be the Galois group of K 2 (2) /K. Our goal is to show that there is some form of d such G is a nonmetacyclic 2-group and give the necessary condition and sufficient for the group G to be metacyclic in case d=2p with p a prime number such that p1(mod4).

DOI : https://doi.org/10.5802/ambp.255
Classification : 11R27,  11R29,  11R37
Mots clés : groupe des unités, système fondamentale d’unités, capitulation, corps de classes de Hilbert, 2-groupe métacyclique
@article{AMBP_2009__16_1_83_0,
     author = {Azizi, Abdelmalek and Taous, Mohammed},
     title = {Condition n\'ecessaire et suffisante pour que certain groupe de Galois soit m\'etacyclique},
     journal = {Annales Math\'ematiques Blaise Pascal},
     pages = {83--92},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {16},
     number = {1},
     year = {2009},
     doi = {10.5802/ambp.255},
     mrnumber = {2514529},
     zbl = {1168.11046},
     language = {fr},
     url = {www.numdam.org/item/AMBP_2009__16_1_83_0/}
}
Azizi, Abdelmalek; Taous, Mohammed. Condition nécessaire et suffisante pour que certain groupe de Galois soit métacyclique. Annales Mathématiques Blaise Pascal, Tome 16 (2009) no. 1, pp. 83-92. doi : 10.5802/ambp.255. http://www.numdam.org/item/AMBP_2009__16_1_83_0/

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