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

Jianing Li; Shenxing Zhang

doi:10.1007/s00209-021-02797-5

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

Number Theory 2021-12-30 v2

Authors: 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 $K$ .

Keywords

3-class group cubic field normal closure Barrucand-Cohn conjecture Lemmermeyer conjecture

Cite

@article{arxiv.2010.05138,
  title  = {The $3$-class groups of $\mathbb{Q}(\sqrt[3]{p})$ and its normal closure},
  author = {Jianing Li and Shenxing Zhang},
  journal= {arXiv preprint arXiv:2010.05138},
  year   = {2021}
}

Comments

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

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