Maximal unramified extensions of imaginary quadratic number fields of small conductors, II
Journal de Théorie des Nombres de Bordeaux, Tome 13 (2001) no. 2, pp. 633-649.

Dans l’article [15], nous donnions dans une table la structure des groupes de Galois Gal(K ur /K) des extensions maximales non ramifiées K ur des corps de nombres quadratiques imaginaires K de conducteur 1000 sous l’Hypothèse de Riemann Généralisée, sauf pour 23 d’entre eux (tous de conducteur 723). Ici nous mettons à jour cette table, en précisant, pour 19 de ces corps exceptionnels, la structure de Gal(K ur /K). En particulier pour K=𝐐(-856), nous obtenons Gal(K ur /K)S 4 ˜×C 5 etK ur =K 4 , le quatrième corps de classes de Hilbert de K. C’est le premier exemple d’un corps de nombres dont la tour de corps de classes est de longueur 4.

In the previous paper [15], we determined the structure of the Galois groups Gal(K ur /K) of the maximal unramified extensions K ur of imaginary quadratic number fields K of conductors 1000 under the Generalized Riemann Hypothesis (GRH) except for 23 fields (these are of conductors 723) and give a table of Gal(K ur /K). We update the table (under GRH). For 19 exceptional fields K of them, we determine Gal(K ur /K). In particular, for K=𝐐(-856), we obtain Gal(K ur /K)S 4 ˜×C 5 andK ur =K 4 , the fourth Hilbert class field of K. This is the first example of a number field whose class field tower has length four.

@article{JTNB_2001__13_2_633_0,
     author = {Yamamura, Ken},
     title = {Maximal unramified extensions of imaginary quadratic number fields of small conductors, II},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {633--649},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {2},
     year = {2001},
     doi = {10.5802/jtnb.341},
     zbl = {1013.11076},
     mrnumber = {1879676},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.341/}
}
Ken Yamamura. Maximal unramified extensions of imaginary quadratic number fields of small conductors, II. Journal de Théorie des Nombres de Bordeaux, Tome 13 (2001) no. 2, pp. 633-649. doi : 10.5802/jtnb.341. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.341/

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