Group theory of cyclic cubic number fields

TL;DR

This paper explores the structure of cyclic cubic number fields sharing a common conductor, revealing new patterns in their class groups and Galois groups through graph-theoretic and algebraic methods.

Contribution

It introduces novel graph-based descriptions of residue conditions that determine the class group structures and Galois groups of these fields, including new classifications of their 3-class towers.

Findings

Identification of bi- and tricyclic 3-class groups

Discovery of metabelian 3-class field towers with coclass ≥ 2

Characterization of groups with order 6561 using abelian invariants

Abstract

Astonishing new discoveries with quartets and octets of cyclic cubic fields sharing a common conductor are presented. Four kinds of graphs describing cubic residue conditions among the prime divisors of the conductor enforce elementary bi- or tricyclic 3-class groups and either a metabelian 3-class field tower group of coclass at least two or a closed Andozhskii-Tsvetkov group of order 6561. In the latter situation, abelian type invariants of first and second order are required for the identification.