New number fields with known p-class tower

TL;DR

This paper investigates the structure of 3-class towers of real quadratic fields, providing a method to identify their Galois groups through subgroup invariants and presenting evidence of their realizations.

Contribution

It introduces a new characterization method for finite 3-groups using abelian quotient invariants and demonstrates their realization as 3-tower groups of real quadratic fields.

Findings

Characterization of finite 3-groups via abelian quotient invariants.

Evidence of specific 3-tower groups in real quadratic fields.

Identification of 3-class tower groups with certain capitulation types.

Abstract

The p-class tower $F_p^\infty(k)$ of a number field k is its maximal unramified pro-p extension. It is considered to be known when the p-tower group, that is the Galois group $G:=Gal(F_p^\infty(k)/k)$, can be identified by an explicit presentation. The main intention of this article is to characterize assigned finite 3-groups uniquely by abelian quotient invariants of subgroups of finite index, and to provide evidence of actual realizations of these groups by 3-tower groups G of real quadratic fields $K=Q(\sqrt{d})$ with 3-capitulation type (0122) or (2034).