Principalization algorithm via class group structure

Daniel C. Mayer[1]

  • [1] Naglergasse 53 8010 Graz Austria

Journal de Théorie des Nombres de Bordeaux (2014)

  • Volume: 26, Issue: 2, page 415-464
  • ISSN: 1246-7405

Abstract

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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 ≤ i ≤ 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 ( 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 ≤ r ≤ 6 , in the sense of Eick, Leedham-Green, and Newman.

How to cite

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Mayer, Daniel C.. "Principalization algorithm via class group structure." Journal de Théorie des Nombres de Bordeaux 26.2 (2014): 415-464. <http://eudml.org/doc/275803>.

@article{Mayer2014,
abstract = {For an algebraic number field $K$ with $3$-class group $\mathrm\{Cl\}_3(K)$ of type $(3,3)$, the structure of the $3$-class groups $\mathrm\{Cl\}_3(N_i)$ of the four unramified cyclic cubic extension fields $N_i$, $1\le i\le 4$, of $K$ is calculated with the aid of presentations for the metabelian Galois group $\mathrm\{G\}_3^2(K)=\mathrm\{Gal\}(\mathrm\{F\}_3^2(K)\vert K)$ of the second Hilbert $3$-class field $\mathrm\{F\}_3^2(K)$ of $K$. In the case of a quadratic base field $K=\mathbb\{Q\}(\sqrt\{D\})$ it is shown that the structure of the $3$-class groups of the four $S_3$-fields $N_1,\ldots ,N_4$ frequently determines the type of principalization of the $3$-class group of $K$ in $N_1,\ldots ,N_4$. 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 $-10^6&lt;D&lt;10^7$ to obtain extensive statistics of their principalization types and the distribution of their second $3$-class groups $\mathrm\{G\}_3^2(K)$ on various coclass trees of the coclass graphs $\mathcal\{G\}(3,r)$, $1\le r\le 6$, in the sense of Eick, Leedham-Green, and Newman.},
affiliation = {Naglergasse 53 8010 Graz Austria},
author = {Mayer, Daniel C.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {$3$-class groups; principalization of $3$-classes; quadratic fields; cubic fields; $S_3$-fields; metabelian $3$-groups; coclass graphs; 3-class groups; principalization of 3-classes; $S_3$ -fields; metabelian 3-groups},
language = {eng},
month = {10},
number = {2},
pages = {415-464},
publisher = {Société Arithmétique de Bordeaux},
title = {Principalization algorithm via class group structure},
url = {http://eudml.org/doc/275803},
volume = {26},
year = {2014},
}

TY - JOUR
AU - Mayer, Daniel C.
TI - Principalization algorithm via class group structure
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/10//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 2
SP - 415
EP - 464
AB - For an algebraic number field $K$ with $3$-class group $\mathrm{Cl}_3(K)$ of type $(3,3)$, the structure of the $3$-class groups $\mathrm{Cl}_3(N_i)$ of the four unramified cyclic cubic extension fields $N_i$, $1\le i\le 4$, of $K$ is calculated with the aid of presentations for the metabelian Galois group $\mathrm{G}_3^2(K)=\mathrm{Gal}(\mathrm{F}_3^2(K)\vert K)$ of the second Hilbert $3$-class field $\mathrm{F}_3^2(K)$ of $K$. In the case of a quadratic base field $K=\mathbb{Q}(\sqrt{D})$ it is shown that the structure of the $3$-class groups of the four $S_3$-fields $N_1,\ldots ,N_4$ frequently determines the type of principalization of the $3$-class group of $K$ in $N_1,\ldots ,N_4$. 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 $-10^6&lt;D&lt;10^7$ to obtain extensive statistics of their principalization types and the distribution of their second $3$-class groups $\mathrm{G}_3^2(K)$ on various coclass trees of the coclass graphs $\mathcal{G}(3,r)$, $1\le r\le 6$, in the sense of Eick, Leedham-Green, and Newman.
LA - eng
KW - $3$-class groups; principalization of $3$-classes; quadratic fields; cubic fields; $S_3$-fields; metabelian $3$-groups; coclass graphs; 3-class groups; principalization of 3-classes; $S_3$ -fields; metabelian 3-groups
UR - http://eudml.org/doc/275803
ER -

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