{
  "id": "2103.04184",
  "version": "v1",
  "published": "2021-03-06T19:58:36.000Z",
  "updated": "2021-03-06T19:58:36.000Z",
  "title": "$3$-Principalization over $S_3$-fields",
  "authors": [
    "Siham Aouissi",
    "Mohamed Talbi",
    "Daniel C. Mayer",
    "Moulay Chrif Ismaili"
  ],
  "comment": "23 pages, 7 Figures, 1 Table",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p\\equiv 1\\,(\\mathrm{mod}\\,9)$ be a prime number and $\\zeta_3$ be a primitive cube root of unity. Then $\\mathrm{k}=\\mathbb{Q}(\\sqrt[3]{p},\\zeta_3)$ is a pure metacyclic field with group $\\mathrm{Gal}(\\mathrm{k}/\\mathbb{Q})\\simeq S_3$. In the case that $\\mathrm{k}$ possesses a $3$-class group $C_{\\mathrm{k},3}$ of type $(9,3)$, the capitulation of $3$-ideal classes of $\\mathrm{k}$ in its unramified cyclic cubic extensions is determined, and conclusions concerning the maximal unramified pro-$3$-extension $\\mathrm{k}_3^{(\\infty)}$ of $\\mathrm{k}$ are drawn.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-06T19:58:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R37",
      "11R29",
      "11R32",
      "11R20",
      "11R16",
      "20D15",
      "20E22",
      "20F05"
    ],
    "keywords": [
      "principalization",
      "pure metacyclic field",
      "unramified cyclic cubic extensions",
      "primitive cube root",
      "prime number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}