Sur le 2-groupe de classes des corps multiquadratiques réels
Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 2, pp. 619-641.

Soient p 1 ,p 2 ,...,p n des nombres premiers distincts -1(mod4), d:=p 1 p 2 p n et k n =Q(p 1 ,p 2 ,...,p n ). On peut approcher le 2-rang du groupe de classes des corps k n en étudiant celui du corps k m (d) pour un entier m<n. Dans cet article, on traite le cas où m=2 ou 3. Comme application, on déduit que le rang du 2-groupe de classes de k 4 est au moins égal à deux (on savait déjà grâce à un résultat de Fröhlich que le groupe de classes de k 4 est toujours d’ordre pair). On en déduit également la liste de tous les corps multiquadratiques k n ayant un 2-groupe de classes cyclique non trivial.

Let p 1 ,p 2 ,...,p n be distinct rational prime numbers -1(mod4), d:=p 1 p 2 p n and k n =Q(p 1 ,p 2 ,...,p n ). The 2-rank of the class group of k n can be approached by studying that of the field k m (d), for an integer m<n. In this article, we treat the case where m=2 or 3. As an application, we deduce that the rank of the 2-class group of k 4 is at least two (according to a result of Fröhlich, we already knew that the class group of k 4 is always of even order). We also draw the list of all multiquadratic fields k n whose 2-class group is cyclic non-trivial.

@article{JTNB_2005__17_2_619_0,
     author = {Ali Mouhib and Abbas Movahhedi},
     title = {Sur le $2$-groupe de classes des corps multiquadratiques r\'eels},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {619--641},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {2},
     year = {2005},
     doi = {10.5802/jtnb.511},
     zbl = {1090.11069},
     mrnumber = {2211311},
     language = {fr},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.511/}
}
Ali Mouhib; Abbas Movahhedi. Sur le $2$-groupe de classes des corps multiquadratiques réels. Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 2, pp. 619-641. doi : 10.5802/jtnb.511. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.511/

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