{
  "id": "2401.12782",
  "version": "v1",
  "published": "2024-01-23T14:11:21.000Z",
  "updated": "2024-01-23T14:11:21.000Z",
  "title": "On indices and monogenity of quartic number fields defined by quadrinomials",
  "authors": [
    "Hamid Ben Yakkou"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Consider a quartic number field $K$ generated by a root of an irreducible quadrinomial of the form $ F(x)= x^4+ax^3+bx+c \\in \\Z[x]$. Let $i(K)$ denote the index of $K$. Engstrom \\cite{Engstrom} established that $i(K)=2^u \\cdot 3^v$ with $u \\le 2$ and $v \\le 1$. In this paper, we provide sufficient conditions on $a$, $b$ and $c$ for $i(K)$ to be divisible by $2$ or $3$, determining the exact corresponding values of $u$ and $v$ in each case. In particular, when $i(K) \\neq 1$, $K$ cannot be monogenic. We also identify new infinite parametric families of monogenic quartic number generated by roots of non-monogenic quadrinomials. We illustrate our results by some computational examples. Our method based on a theorem of Ore on the decomposition of primes in number fields \\cite{Nar,O}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-23T14:11:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R04",
      "11R16",
      "11R21",
      "11Y40",
      "F.2.2"
    ],
    "keywords": [
      "quartic number field",
      "monogenity",
      "infinite parametric families",
      "monogenic quartic number",
      "sufficient conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}