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

Abstract

We determine the $3$-class groups of $\mathbb{Q}(\sqrt[3]{p})$ and $K=\mathbb{Q}(\sqrt[3]{p},\sqrt{-3})$ when $p\equiv 4,7\mathrm{mod}9$ 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 $K$.

Add Schoof's remark in the end. (to appear in Math. Z.)

References in corpus (2)

• $\ell$-Class groups of fields in Kummer towers
• On a Conjecture of Lemmermeyer