Maximal unramified extensions of imaginary quadratic number fields of small conductors, II

Yamamura, Ken

Yamamura, Ken

Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 2, pp. 633-649.

Abstract
Résumé

In the previous paper [15], we determined the structure of the Galois groups $Gal (K_{u r} / K)$ of the maximal unramified extensions $K_{u r}$ 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 $Gal (K_{u r} / K)$ . We update the table (under GRH). For 19 exceptional fields $K$ of them, we determine $Gal (K_{u r} / K)$ . In particular, for $K = 𝐐 (\sqrt{- 856})$ , we obtain $Gal (K_{u r} / K) ≅ \tilde{S_{4}} \times C_{5} and K_{u r} = K_{4}$ , 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 $Gal (K_{u r} / K)$ des extensions maximales non ramifiées $K_{u r}$ 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 $Gal (K_{u r} / K)$ . En particulier pour $K = 𝐐 (\sqrt{- 856})$ , nous obtenons $Gal (K_{u r} / K) ≅ \tilde{S_{4}} \times C_{5} et K_{u r} = K_{4}$ , 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 $4$ .

MR Zbl

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     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},
     mrnumber = {1879676},
     zbl = {1013.11076},
     language = {en},
     url = {http://archive.numdam.org/item/JTNB_2001__13_2_633_0/}
}

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Yamamura, Ken. 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. http://archive.numdam.org/item/JTNB_2001__13_2_633_0/

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