{
  "id": "1911.11198",
  "version": "v1",
  "published": "2019-11-25T20:03:26.000Z",
  "updated": "2019-11-25T20:03:26.000Z",
  "title": "On the $2$-class group of some number fields with large degree",
  "authors": [
    "Mohamed Mahmoud Chems-Eddin",
    "Abdelmalek Azizi",
    "Abdelkader Zekhnini"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $d$ be an odd square-free integer, $m\\geq 3$, $k$:$=\\mathbb{Q}(\\sqrt{d}, \\sqrt{-1})$, $\\mathbb{Q}(\\sqrt{-2}, \\sqrt{d})$ or $\\mathbb{Q}(\\sqrt{-2}, \\sqrt{-d})$, and $L_{m,d}:=\\mathbb{Q}(\\zeta_{2^m},\\sqrt{d})$. In this paper, we shall determine all the fields $L_{m, d}:=\\mathbb{Q}(\\zeta_{2^m},\\sqrt{d})$, with $m\\geq 3$ is an integer, such that the class number of $L_{m, d}$ is odd. Furthermore, using the cyclotomic $\\mathbb{Z}_2$-extensions of $k$, we compute the rank of the $2$-class group of $L_{m, d}$ whenever the divisors of $d$ are congruent $3$ or $5\\pmod 8$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-25T20:03:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "class group",
      "number fields",
      "large degree",
      "odd square-free integer",
      "class number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}