{
  "id": "1404.3761",
  "version": "v1",
  "published": "2014-04-14T21:37:07.000Z",
  "updated": "2014-04-14T21:37:07.000Z",
  "title": "Principalization of $2$-class groups of type $(2,2,2)$ of biquadratic fields $\\mathbb{Q}\\left(\\sqrt{\\strut p_1p_2q},\\sqrt{\\strut -1}\\right)$",
  "authors": [
    "Abdelmalek Azizi",
    "Abdelkader Zekhnini",
    "Mohammed Taous",
    "Daniel C. Mayer"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p_1\\equiv p_2\\equiv -q\\equiv1 \\pmod4$ be different primes such that $\\displaystyle\\left(\\frac{2}{p_1}\\right)= \\displaystyle\\left(\\frac{2}{p_2}\\right)=\\displaystyle\\left(\\frac{p_1}{q}\\right)=\\displaystyle\\left(\\frac{p_2}{q}\\right)=-1$. Put $d=p_1p_2q$ and $i=\\sqrt{-1}$, then the bicyclic biquadratic field ${k}=\\mathbb{Q}(\\sqrt{d},i)$ has an elementary abelian $2$-class group, $\\mathbf{C}l_2(k)$, of rank $3$. In this paper, we study the principalization of the $2$-classes of ${k}$ in its fourteen unramified abelian extensions $\\mathbb{K}_j$ and $\\mathbb{L}_j$ within ${k}_2^{(1)}$, that is the Hilbert $2$-class field of ${k}$. We determine the nilpotency class, the coclass, generators and the structure of the metabelian Galois group $G=\\mathrm{Gal}(\\mathbb{L}/{k})$ of the second Hilbert 2-class field ${k}_2^{(2)}$ of ${k}$. Additionally, the abelian type invariants of the groups $\\mathbf{C}l_2(\\mathbb{K}_j)$ and $\\mathbf{C}l_2(\\mathbb{L}_j)$ and the length of the $2$-class tower of ${k}$ are given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-14T21:37:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R16",
      "11R29",
      "11R32",
      "11R37",
      "20D15"
    ],
    "keywords": [
      "class group",
      "principalization",
      "bicyclic biquadratic field",
      "metabelian galois group",
      "abelian type invariants"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.3761A"
    }
  }
}