{
  "id": "2010.05138",
  "version": "v1",
  "published": "2020-10-11T02:16:47.000Z",
  "updated": "2020-10-11T02:16:47.000Z",
  "title": "The $3$-class groups of $\\mathbb{Q}(\\sqrt[3]{p})$ and its normal closure",
  "authors": [
    "Jianing Li",
    "Shenxing Zhang"
  ],
  "comment": "6 pages, comments welcome",
  "categories": [
    "math.NT"
  ],
  "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 $K$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-11T02:16:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R29",
      "11R16"
    ],
    "keywords": [
      "class group",
      "normal closure",
      "cubic modulo",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}