Index-\(p\) abelianization data of \(p\)-class tower groups

TL;DR

This paper studies the abelian invariants of p-class groups of number fields to understand their Galois groups and unramified extensions, providing complete classifications for certain cases and methods for identifying maximal unramified pro-p groups.

Contribution

It offers a complete classification of index- extit{p} abelianization data for p=3 with rank-two class groups and introduces iterated IPADs to identify maximal unramified pro- extit{p} groups.

Findings

Complete list of IPADs for p=3, rank-two class groups.

Method to determine the Galois group of second Hilbert p-class field.

Use of iterated IPADs to identify maximal unramified pro-p extensions.

Abstract

Given a fixed prime number \(p\), the multiplet of abelian type invariants of the \(p\)-class groups of all unramified cyclic degree \(p\) extensions of a number field \(K\) is called its IPAD (index-\(p\) abelianization data). These invariants have proved to be a valuable information for determining the Galois group \(G_p^2\) of the second Hilbert \(p\)-class field and the \(p\)-capitulation type \(\varkappa\) of \(K\). For \(p=3\) and a number field \(K\) with elementary \(p\)-class group of rank two, all possible IPADs are given in the complete form of several infinite sequences. Iterated IPADs of second order are used to identify the group \(G_p^\infty\) of the maximal unramified pro-\(p\) extension of \(K\).