The distribution of second p -class groups on coclass graphs

Daniel C. Mayer[1]

  • [1] Naglergasse 53 8010 Graz Austria

Journal de Théorie des Nombres de Bordeaux (2013)

  • Volume: 25, Issue: 2, page 401-456
  • ISSN: 1246-7405

Abstract

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General concepts and strategies are developed for identifying the isomorphism type of the second p -class group G = Gal ( F p 2 ( K ) | K ) , that is the Galois group of the second Hilbert p -class field F p 2 ( K ) , of a number field K , for a prime p . The isomorphism type determines the position of G on one of the coclass graphs 𝒢 ( p , r ) , r ≥ 0 , in the sense of Eick, Leedham-Green, and Newman. It is shown that, for special types of the base field K and of its p -class group Cl p ( K ) , the position of G is restricted to certain admissible branches of coclass trees by selection rules. Deeper insight, in particular, the density of population of individual vertices on coclass graphs, is gained by computing the actual distribution of second p -class groups G for various series of number fields K having p -class groups Cl p ( K ) of fixed type and p ∈ { 2 , 3 , 5 , 7 } .

How to cite

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Mayer, Daniel C.. "The distribution of second $p$-class groups on coclass graphs." Journal de Théorie des Nombres de Bordeaux 25.2 (2013): 401-456. <http://eudml.org/doc/275771>.

@article{Mayer2013,
abstract = {General concepts and strategies are developed for identifying the isomorphism type of the second $p$-class group $G=\mathrm\{Gal\}(\mathrm\{F\}_p^2(K)\vert K)$, that is the Galois group of the second Hilbert $p$-class field $\mathrm\{F\}_p^2(K)$, of a number field $K$, for a prime $p$. The isomorphism type determines the position of $G$ on one of the coclass graphs $\mathcal\{G\}(p,r)$, $r\ge 0$, in the sense of Eick, Leedham-Green, and Newman. It is shown that, for special types of the base field $K$ and of its $p$-class group $\mathrm\{Cl\}_p(K)$, the position of $G$ is restricted to certain admissible branches of coclass trees by selection rules. Deeper insight, in particular, the density of population of individual vertices on coclass graphs, is gained by computing the actual distribution of second $p$-class groups $G$ for various series of number fields $K$ having $p$-class groups $\mathrm\{Cl\}_p(K)$ of fixed type and $p\in \lbrace 2,3,5,7\rbrace $.},
affiliation = {Naglergasse 53 8010 Graz Austria},
author = {Mayer, Daniel C.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {$p$-class groups; $p$-class field tower; principalization of $p$-classes; quadratic fields; cubic fields; dihedral fields; metabelian $p$-groups; coclass graphs; -class group; -class field tower; principalization of -classes; quadratic field; cubic field; metabelian -group; coclass graph},
language = {eng},
month = {9},
number = {2},
pages = {401-456},
publisher = {Société Arithmétique de Bordeaux},
title = {The distribution of second $p$-class groups on coclass graphs},
url = {http://eudml.org/doc/275771},
volume = {25},
year = {2013},
}

TY - JOUR
AU - Mayer, Daniel C.
TI - The distribution of second $p$-class groups on coclass graphs
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2013/9//
PB - Société Arithmétique de Bordeaux
VL - 25
IS - 2
SP - 401
EP - 456
AB - General concepts and strategies are developed for identifying the isomorphism type of the second $p$-class group $G=\mathrm{Gal}(\mathrm{F}_p^2(K)\vert K)$, that is the Galois group of the second Hilbert $p$-class field $\mathrm{F}_p^2(K)$, of a number field $K$, for a prime $p$. The isomorphism type determines the position of $G$ on one of the coclass graphs $\mathcal{G}(p,r)$, $r\ge 0$, in the sense of Eick, Leedham-Green, and Newman. It is shown that, for special types of the base field $K$ and of its $p$-class group $\mathrm{Cl}_p(K)$, the position of $G$ is restricted to certain admissible branches of coclass trees by selection rules. Deeper insight, in particular, the density of population of individual vertices on coclass graphs, is gained by computing the actual distribution of second $p$-class groups $G$ for various series of number fields $K$ having $p$-class groups $\mathrm{Cl}_p(K)$ of fixed type and $p\in \lbrace 2,3,5,7\rbrace $.
LA - eng
KW - $p$-class groups; $p$-class field tower; principalization of $p$-classes; quadratic fields; cubic fields; dihedral fields; metabelian $p$-groups; coclass graphs; -class group; -class field tower; principalization of -classes; quadratic field; cubic field; metabelian -group; coclass graph
UR - http://eudml.org/doc/275771
ER -

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