Group theoretic approach to cyclic cubic fields

TL;DR

This paper investigates the structure of certain cyclic cubic number fields sharing a common conductor, determining their automorphism groups and maximal unramified extensions using class group properties and residue symbols.

Contribution

It introduces a group theoretic framework to analyze cyclic cubic fields with shared conductors, linking automorphism groups to residue symbols and principal ideals.

Findings

Automorphism group M characterized by residue symbols and principal ideals.

Conditions under which M equals the Galois group G of the maximal unramified pro-3-extension.

Explicit criteria for the structure of 3-class groups in these fields.

Abstract

Let (k1,k2,k3,k4) be a quartet of cyclic cubic number fields sharing a common conductor c=pqr divisible by exactly three prime(power)s p,q,r. For those components k of the quartet whose 3-class group Cl(3,k) = Z/3Z x Z/3Z is elementary bicyclic, the automorphism group M = Gal(F(3,2,k)/k) of the maximal metabelian unramified 3-extension of k is determined by conditions for cubic residue symbols between p,q,r and for ambiguous principal ideals in subfields of the common absolute 3-genus field k* of k1,k2,k3,k4. With the aid of the relation rank d2(M), it is decided whether M coincides with the Galois group G = Gal(F(3,infinity,k)/k) of the maximal unramified pro-3-extension of k.