The generators of $3$-class group of some fields of degree $6$ over $\mathbb{Q}$

TL;DR

This paper determines explicit generators for the 3-part of the class group of certain degree 6 fields over Q, specifically when the group is isomorphic to Z/9Z x Z/3Z, building on previous conjecture proofs.

Contribution

It explicitly identifies generators of the 3-class group for specific degree 6 fields, extending prior theoretical results.

Findings

Generators of the 3-class group are explicitly constructed.

Conditions for the 3-class group to be Z/9Z x Z/3Z are characterized.

The work confirms the structure of the 3-class group in these cases.

Abstract

Let $\mathrm{k}=\mathbb{Q}\left(\sqrt[3]{p},\zeta_3\right)$, where $p$ is a prime number such that $p \equiv 1 \pmod 9$, and let $C_{\mathrm{k},3}$ be the $3$-component of the class group of $\mathrm{k}$. In \cite{GERTH3}, Frank Gerth III proves a conjecture made by Calegari and Emerton \cite{Cal-Emer} which gives necessary and sufficient conditions for $C_{\mathrm{k},3}$ to be of $\operatorname{rank}\,$ two. The purpose of the present work is to determine generators of $C_{\mathrm{k},3}$, whenever it is isomorphic to $\mathbb{Z}/9\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$.