On a Conjecture of Lemmermeyer

TL;DR

This paper proves Lemmermeyer's conjecture concerning the structure of 3-class groups in pure cubic fields and their normal closures, advancing understanding in algebraic number theory.

Contribution

It provides a proof of Lemmermeyer's conjecture about 3-class groups in pure cubic fields and their normal closures.

Findings

Confirmed the conjecture for primes p ≡ 1 mod 3

Established new properties of 3-class groups in pure cubic fields

Enhanced understanding of class group structures in algebraic number theory

Abstract

Let $p\equiv 1\,(\mathrm{mod}\,3)$ be a prime and denote by $\zeta_3$ a primitive third root of unity. Recently, Lemmermeyer presented a conjecture about $3$-class groups of pure cubic fields $L=\mathbb{Q}(\sqrt[3]{p})$ and of their normal closures $\mathrm{k}=\mathbb{Q}(\sqrt[3]{p},\zeta_3)$. The purpose of this paper is to prove Lemmermeyer's conjecture.