{
  "id": "1403.0662",
  "version": "v1",
  "published": "2014-03-04T02:19:31.000Z",
  "updated": "2014-03-04T02:19:31.000Z",
  "title": "On the rank of the $2$-class group of $\\mathbb{Q}(\\sqrt{p}, \\sqrt{q},\\sqrt{-1})$",
  "authors": [
    "Abdelmalek Azizi Mohammed Taous",
    "Abdelkader Zekhnini"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $d$ be a square-free integer, $\\mathbf{k}=\\mathbb{Q}(\\sqrt d,\\,i)$ and $i=\\sqrt{-1}$. Let $\\mathbf{k}_1^{(2)}$ be the Hilbert $2$-class field of $\\mathbf{k}$, $\\mathbf{k}_2^{(2)}$ be the Hilbert $2$-class field of $\\mathbf{k}_1^{(2)}$ and $G=\\mathrm{Gal}(\\mathbf{k}_2^{(2)}/\\mathbf{k})$ be the Galois group of $\\mathbf{k}_2^{(2)}/\\mathbf{k}$. Our goal is to give necessary and sufficient conditions to have $G$ metacyclic in the case where $d=pq$, with $p$ and $q$ are primes such that $p\\equiv 1\\pmod 8$ and $q\\equiv 5\\pmod 8$ or $p\\equiv 1\\pmod 8$ and $q\\equiv 3\\pmod 4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-04T02:19:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R11",
      "11R29",
      "11R32",
      "11R37"
    ],
    "keywords": [
      "class group",
      "class field",
      "square-free integer",
      "galois group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.0662T"
    }
  }
}