Criteria for Three-Stage Towers of &amp;lt;i&amp;gt;p&amp;lt;/i&amp;gt;-Class Fields

Daniel C Mayer

doi:10.4236/apm.2017.72008

Abstract

Let p be a prime and K be a number field with non-trivial p-class group ClpK. A crucial step in identifying the Galois group G∞p of the maximal unramified pro-p extension of K is to determine its two-stage approximation M=G2pk, that is the second derived quotient MsG/Gn. The family τ1K of abelian type invariants of the p-class groups ClpL of all unramified cyclic extensions L/K of degree p is called the index- abelianization data (IPAD) of K. It is able to specify a finite batch of contestants for the second p-class group M of K. In this paper we introduce two different kinds of generalized IPADs for obtaining more sophisticated results. The multi-layered IPAD (τ1Kτ(2)K) includes data on unramified abelian extensions L/K of degree p2 and enables sharper bounds for the order of M in the case Clpks(p,p,p), where current im-plementations of the p-group generation algorithm fail to produce explicit contestants for M , due to memory limitations. The iterated IPAD of second order τ(2)K contains information on non-abelian unramified extensions L/K of degree p2, or even p3, and admits the identification of the p-class tower group G for various infinite series of quadratic fields K=Q(√d) with ClpKs(p,p) possessing a p-class field tower of exact length lpK=3 as a striking novelty.

Highlights

This is the unique situation where all index- p abelianization data (IPAD) can be given in a complete form, except for the simple case of a number field K with 2-class group Cl2K of type (2,2) [[5], § 9, pp. 501-503]
In §§7.5, 7.8, we provide evidence of unexpected phenomena revealed by real quadratic fields K with types 1K in Scholz and Taussky’s section E [[6], p. 36]
According to the Artin reciprocity law of class field theory [9], the unramified cyclic extensions L K of relative degree p of a number field K with non-trivial p -class group Clp K are in a bijective correspondence to the subgroups of index p in Cl p K

Summary

Introduction

290-291] in a more succinct and elegant form avoiding infinitely many exceptions, and emphasizing the role of two distinguished components, called the polarization and co-polarization, which are crucial for proving the finiteness of the batch of contestants for the second 3-class group M = G32 K. Up to now, this is the unique situation where all IPADs can be given in a complete form, except for the simple case of a number field K with 2-class group Cl2K of type (2,2) [[5], § 9, pp.

Index- p Abelianization Data

The p -Principalization Type

The Artin Transfer Pattern