On a Conjecture of Lemmermeyer

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.

Pure cubic fields, Galois closure, $3$-class groups, abelian type invariants

Cited by in corpus (2)

• $\ell$-Class groups of fields in Kummer towers
• The $3$-class groups of $\mathbb{Q}(\sqrt[3]{p})$ and its normal closure