Principalization algorithm via class group structure
Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 415-464.

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 , 1i4, 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=(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 4596 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), 1r6, 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 , 1i4, 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=(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 4596 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), 1r6, in the sense of Eick, Leedham-Green, and Newman.

Reçu le : 2011-08-29
Révisé le : 2013-04-25
Accepté le : 2014-03-13
Publié le : 2015-03-09
DOI : https://doi.org/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
@article{JTNB_2014__26_2_415_0,
     author = {Daniel C. Mayer},
     title = {Principalization algorithm via class group structure},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {415--464},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {26},
     number = {2},
     year = {2014},
     doi = {10.5802/jtnb.874},
     mrnumber = {3320487},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2014__26_2_415_0/}
}
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/item/JTNB_2014__26_2_415_0/

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