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arXiv · 2010.05138

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

Jianing Li·Shenxing Zhang
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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 $K$.

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Jianing Li, Shenxing Zhang. 2020-10-11. The $3$-class groups of $\mathbb{Q}(\sqrt[3]{p})$ and its normal closure. https://doi.org/10.1007/s00209-021-02797-5

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