{
  "id": "2206.13931",
  "version": "v1",
  "published": "2022-06-28T12:05:55.000Z",
  "updated": "2022-06-28T12:05:55.000Z",
  "title": "Unlimited lists of fundamental units of quadratic fields -- Applications",
  "authors": [
    "Georges Gras"
  ],
  "comment": "26 pages. Elementary subject about real quadratic fields with new algorithms using given PARI programs",
  "categories": [
    "math.NT"
  ],
  "abstract": "We use the polynomials $m_s(t) = t^2 - 4 s$, $s \\in \\{-1, 1\\}$, in an elementary process giving arbitrary large lists of {\\it fundamental units} of quadratic fields of discriminants listed in ascending order. More precisely, let $\\mathbf{B} \\gg 0$; then as $t$ grows from $1$ to $\\mathbf{B}$, for each {\\it first occurrence} of a square-free integer $M \\geq 2$, in the factorization $m_s(t) =: M r^2$, the unit $\\frac{1}{2} \\big(t + r \\sqrt{M}\\big)$ is the fundamental unit of norm $s$ of $\\mathbb{Q}(\\sqrt M)$, even if $r >1$ (Theorem 4.1). Using $m_{s\\nu}(t) = t^2 - 4 s \\nu$, $\\nu \\geq 2$, the algorithm gives arbitrary large lists of {\\it fundamental solutions} to $u^2 - M v^2= 4s\\nu$ (Theorem 4.11). We deduce, for $p>2$ prime, arbitrary large lists of {\\it non $p$-rational} quadratic fields (Theorems 6.3, 6.4, 6.5) and of degree $p-1$ imaginary fields with non-trivial $p$-class group (Theorems 7.1,7.2). PARI programs are given to be copied and pasted.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-28T12:05:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fundamental unit",
      "quadratic fields",
      "unlimited lists",
      "process giving arbitrary large lists",
      "elementary process giving arbitrary large"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}