{
  "id": "1805.04963",
  "version": "v1",
  "published": "2018-05-13T22:04:41.000Z",
  "updated": "2018-05-13T22:04:41.000Z",
  "title": "Fields $\\mathbb{Q}(\\sqrt[3]{d},ζ_3)$ whose $3$-class group is of type $(9,3)$",
  "authors": [
    "Siham Aouissi",
    "Mohamed Talbi",
    "Moulay Chrif Ismaili",
    "Abdelmalek Azizi"
  ],
  "comment": "10 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\mathrm{k}=\\mathbb{Q}(\\sqrt[3]{d},\\zeta_3)$, with $d$ a cube-free positive integer. Let $C_{\\mathrm{k},3}$ be the $3$-component of the class group of $\\mathrm{k}$. By the aid of genus theory, arithmetic proprieties of the pure cubic field $\\mathbb{Q}(\\sqrt[3]{d})$ and some results on the $3$-class group $C_{\\mathrm{k},3}$, we are moving towards the determination of all integers $d$ such that $C_{\\mathrm{k},3} \\simeq \\mathbb{Z}/9\\mathbb{Z}\\times \\mathbb{Z}/3\\mathbb{Z}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-13T22:04:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R11",
      "11R16",
      "11R20",
      "11R27",
      "11R29",
      "11R37"
    ],
    "keywords": [
      "class group",
      "pure cubic field",
      "cube-free positive integer",
      "genus theory",
      "arithmetic proprieties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}