{
  "id": "2408.14117",
  "version": "v1",
  "published": "2024-08-26T09:06:04.000Z",
  "updated": "2024-08-26T09:06:04.000Z",
  "title": "On characterization of Monogenic number fields associated with certain quadrinomials and its applications",
  "authors": [
    "Tapas Chatterjee",
    "Karishan Kumar"
  ],
  "comment": "23 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f(x)=x^{n}+ax^{3}+bx+c$ be the minimal polynomial of an algebraic integer $\\theta$ over the rationals with certain conditions on $a,~b,~c,$ and $n.$ Let $K=\\mathbb{Q}(\\theta)$ be a number field and $\\mathcal{O}_{K}$ be the ring of integers of $K.$ In this article, we characterize all the prime divisors of the discriminant of $f(x)$ which do not divide the index of $\\theta.$ As an interesting result, we establish necessary and sufficient conditions for the field $K=\\mathbb{Q}(\\theta)$ to be monogenic. Finally, we investigate the types of solutions to certain differential equations associated with the polynomial $f(x).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-26T09:06:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R04",
      "11R29",
      "11Y40",
      "11R09",
      "11R21"
    ],
    "keywords": [
      "monogenic number fields",
      "characterization",
      "quadrinomials",
      "applications",
      "minimal polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}