{
  "id": "2412.20443",
  "version": "v1",
  "published": "2024-12-29T11:54:55.000Z",
  "updated": "2024-12-29T11:54:55.000Z",
  "title": "Monogenic $S_3$-cubic polynomials and class numbers of associated quadratic fields",
  "authors": [
    "Lenny Jones"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We say that a monic polynomial $f(x)\\in {\\mathbb Z}[x]$ is monogenic if $f(x)$ is irreducible over ${\\mathbb Q}$ and $\\{1,\\theta,\\theta^2,\\ldots ,\\theta^{\\deg{f}-1}\\}$ is a basis for the ring of integers of ${\\mathbb Q}(\\theta)$, where $f(\\theta)=0$. In this article, we use a theorem due to Kishi and Miyake to give an explicit description of an infinite set ${\\mathcal F}$ of cubic polynomials $f(x)\\in {\\mathbb Z}[x]$, such that $f(x)$ satisfies the three conditions: ${\\rm Gal}_{{\\mathbb Q}}(f)\\simeq S_3$, the class number of the unique quadratic subfield of the splitting field of $f(x)$ is divisible by 3, and $f(x)$ is monogenic. Moreover, if $g(x)$ is any cubic polynomial satisfying these three conditions, then there exists a unique $f(x)\\in {\\mathcal F}$ such that $f(x)$ and $g(x)$ generate the same cubic extension over ${\\mathbb Q}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-29T11:54:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cubic polynomial",
      "associated quadratic fields",
      "class number",
      "unique quadratic subfield",
      "monic polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}