Principalization algorithm via class group structure

Daniel C. Mayer

doi:10.5802/jtnb.874

Daniel C. Mayer ¹

¹ Naglergasse 53 8010 Graz Austria

Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 415-464.

Résumé
Abstract

Pour un corps de nombres algébriques $K$ avec $3$ -groupe de classes ${Cl}_{3} (K)$ du type $(3, 3)$ , la structure des $3$ -groupes de classes ${Cl}_{3} (N_{i})$ des quatre extensions cubiques cycliques non-ramifiées $N_{i}$ , $1 \leq i \leq 4$ , de $K$ est calculée à l’aide de présentations du groupe de Galois métabélien $G_{3}^{2} (K) = Gal (F_{3}^{2} (K) | K)$ du deuxième $3$ -corps de classes de Hilbert $F_{3}^{2} (K)$ de $K$ . Dans le cas d’un corps de base quadratique $K = ℚ (\sqrt{D})$ , il est montré que la structure des $3$ -groupes de classes des quatre $S_{3}$ -corps $N_{1}, ..., N_{4}$ détermine fréquemment le type de principalisation de $3$ -groupe de classes de $K$ dans $N_{1}, ..., N_{4}$ . Ceci offre une alternative à l’algorithme de principalisation classique par Scholz et Taussky. Le nouvel algorithme qui est facilement automatisable et s’exécute très brièvement est implémenté en PARI/GP et est appliqué à tous les $4 596$ corps quadratiques $K$ de $3$ -groupe de classes du type $(3, 3)$ et de discriminant $- 10^{6} < D < 10^{7}$ pour obtenir des statistiques détaillées sur leurs types de principalisation et sur la distribution de leurs deuxièmes $3$ -groupes de classes $G_{3}^{2} (K)$ sur des arbres divers de coclasses des graphes de coclasses $𝒢 (3, r)$ , $1 \leq r \leq 6$ , dans le sens de Eick, Leedham-Green et Newman.

For an algebraic number field $K$ with $3$ -class group ${Cl}_{3} (K)$ of type $(3, 3)$ , the structure of the $3$ -class groups ${Cl}_{3} (N_{i})$ of the four unramified cyclic cubic extension fields $N_{i}$ , $1 \leq i \leq 4$ , of $K$ is calculated with the aid of presentations for the metabelian Galois group $G_{3}^{2} (K) = Gal (F_{3}^{2} (K) | K)$ of the second Hilbert $3$ -class field $F_{3}^{2} (K)$ of $K$ . In the case of a quadratic base field $K = ℚ (\sqrt{D})$ it is shown that the structure of the $3$ -class groups of the four $S_{3}$ -fields $N_{1}, ..., N_{4}$ frequently determines the type of principalization of the $3$ -class group of $K$ in $N_{1}, ..., N_{4}$ . This provides an alternative to the classical principalization algorithm by Scholz and Taussky. The new algorithm, which is easily automatizable and executes very quickly, is implemented in PARI/GP and is applied to all $4 596$ quadratic fields $K$ with $3$ -class group of type $(3, 3)$ and discriminant $- 10^{6} < D < 10^{7}$ to obtain extensive statistics of their principalization types and the distribution of their second $3$ -class groups $G_{3}^{2} (K)$ on various coclass trees of the coclass graphs $𝒢 (3, r)$ , $1 \leq r \leq 6$ , in the sense of Eick, Leedham-Green, and Newman.

MR

DOI : 10.5802/jtnb.874

Classification : 11R29, 11R11, 11R16, 11R20, 20D15
Mots clés : $3$-class groups, principalization of $3$-classes, quadratic fields, cubic fields, $S_3$-fields, metabelian $3$-groups, coclass graphs

Affiliations des auteurs :

Daniel C. Mayer ¹

¹ Naglergasse 53 8010 Graz Austria

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Daniel C. Mayer. Principalization algorithm via class group structure. Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 415-464. doi : 10.5802/jtnb.874. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.874/

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