The $3$-class groups of $\mathbb{Q}(\sqrt[3]{p})$ and its normal closure

TL;DR

This paper determines the structure of the 3-class groups of certain cubic fields and their normal closures, confirming longstanding conjectures and completing the classification for specific prime cases.

Contribution

It explicitly computes the 3-class groups of ield and its normal closure for primes with specific congruence conditions, confirming and completing previous conjectures.

Findings

Confirmed Barrucand-Cohn conjecture for these fields.

Proved the last remaining case of Lemmermeyer's conjecture on 3-class groups.

Established the structure of 3-class groups for primes p or which 3 is cubic mod p.

Abstract

We determine the $3$-class groups of $\mathbb{Q}(\sqrt[3]{p})$ and $K=\mathbb{Q}(\sqrt[3]{p},\sqrt{-3})$ when $p\equiv 4,7\bmod 9$ is a prime and $3$ is a cubic modulo $p$. This confirms a conjecture made by Barrucand-Cohn, and proves the last remaining case of a conjecture of Lemmermeyer on the $3$-class group of $K$.