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

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.

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.

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Ken Yamamura. Maximal unramified extensions of imaginary quadratic number fields of small conductors, II. Journal de Théorie des Nombres de Bordeaux, Volume 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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