{
  "id": "2204.02436",
  "version": "v1",
  "published": "2022-04-05T18:27:46.000Z",
  "updated": "2022-04-05T18:27:46.000Z",
  "title": "On monogenity of certain pure number fields defined by $x^{2^u\\cdot 3^v\\cdot 5^t}-m$",
  "authors": [
    "Lhoussain El Fadil"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:2106.01252, arXiv:2112.01133, arXiv:2111.05899, arXiv:2106.00004; text overlap with arXiv:2203.13353",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $K = \\mathbb{Q} (\\alpha) $ be a pure number field generated by a root $\\alpha$ of a monic irreducible polynomial $ F(x) = x^{2^u\\cdot 3^v\\cdot 5^t}-m$, with $ m \\neq \\pm 1 $ a square free rational integer, $u$, $v$ and $t$ three positive integers. In this paper, we study the monogenity of $K$. We prove that if $m\\not\\equiv 1\\md4$, $m\\not\\equiv \\pm 1\\md9$, and $m\\not\\in\\{\\pm 1, \\pm 7\\}\\md{25}$, then $K$ is monogenic. But if {$m\\equiv 1\\md{4}$} or $m\\equiv 1\\md9$ or $m\\equiv -1\\md9$ and $u=2k$ for some odd integer $k$ or $u\\ge 2$ and $m\\equiv 1\\md{25}$ or $m\\equiv -1\\md{25}$ and $u=2k$ for some odd integer $k$ or $u=v=1$ and $m\\equiv \\pm 82\\md{5^4}$, then $K$ is not monogenic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-05T18:27:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R04",
      "11R16",
      "11R21",
      "D.2"
    ],
    "keywords": [
      "pure number field",
      "monogenity",
      "square free rational integer",
      "odd integer",
      "monic irreducible polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}