Maximal unramified extensions of imaginary quadratic number fields of small conductors

Ken Yamamura

Ken Yamamura

Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 2, pp. 405-448.

Résumé
Abstract

Nous déterminons la structure du groupe de Galois Gal $(K_{u r} / K)$ de l’extension maximale non ramifiée $K_{u r}$ de chaque corps quadratique imaginaire de conducteur $≦ 420 (≦ 719$ sous GRH). Pour tous ces corps $K$ , l’extension $K_{u r}$ coïncide avec $K$ , ou avec le corps de classes de Hilbert de $K$ , ou avec le second corps de classes de Hilbert de $K$ ou avec le troisième corps de classes de Hilbert de $K$ . Les bornes d’Odlyzko sur les discriminants et les informations sur la structure des groupes de classes obtenues par l’action du groupe de Galois sur les groupes de classes sont ici essentielles. Nous utilisons aussi des relations sur le nombre de classes et un ordinateur pour le calcul du nombre de classes de corps de bas degré pour obtenir le nombre de classes de corps de degré plus élevé. Nous utilisons aussi des résultats sur les tours de corps de classes, ainsi que notre connaissance des $2$ -groupes d’ordre $≦ 2^{6}$ et des groupes linéaires sur des corps finis.

We determine the structures 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 $≦ 420 (≦ 719$ under the Generalized Riemann Hypothesis). For all such $K$ , $K_{u r}$ is $K$ , the Hilbert class field of $K$ , the second Hilbert class field of $K$ , or the third Hilbert class field of $K$ . The use of Odlyzko’s discriminant bounds and information on the structure of class groups obtained by using the action of Galois groups on class groups is essential. We also use class number relations and a computer for calculation of class numbers of fields of low degrees in order to get class numbers of fields of higher degrees. Results on class field towers and the knowledge of the $2$ -groups of orders $≦ 2^{6}$ and linear groups over finite fields are also used.

MR Zbl

Classification : Primary 11R32, 11R11
Mots-clés : maximal unramified extension, imaginary quadratic number field, discriminant bounds, class field tower

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     pages = {405--448},
     publisher = {Universit\'e Bordeaux I},
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     year = {1997},
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     mrnumber = {1617407},
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     url = {https://jtnb.centre-mersenne.org/item/JTNB_1997__9_2_405_0/}
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Ken Yamamura. Maximal unramified extensions of imaginary quadratic number fields of small conductors. Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 2, pp. 405-448. https://jtnb.centre-mersenne.org/item/JTNB_1997__9_2_405_0/

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