{
  "id": "0811.1300",
  "version": "v1",
  "published": "2008-11-08T22:48:01.000Z",
  "updated": "2008-11-08T22:48:01.000Z",
  "title": "On Quadratic Fields Generated by Discriminants of Irreducible Trinomials",
  "authors": [
    "I. E. Shparlinski"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "A. Mukhopadhyay, M. R. Murty and K. Srinivas (http://arxiv.org/abs/0808.0418) have recently studied various arithmetic properties of the discriminant $\\Delta_n(a,b)$ of the trinomial $f_{n,a,b}(t) = t^n + at + b$, where $n \\ge 5$ is a fixed integer. In particular, it is shown that, under the $abc$-conjecture, for every $n \\equiv 1 \\pmod 4$, the quadratic fields $\\Q(\\sqrt{\\Delta_n(a,b)})$ are pairwise distinct for a positive proportion of such discriminants with integers $a$ and $b$ such that $f_{n,a,b}$ is irreducible over $\\Q$ and $|\\Delta_n(a,b)|\\le X$, as $X\\to \\infty$. We use the square-sieve and bounds of character sums to obtain a weaker but unconditional version of this result.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-11-08T22:48:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11L40",
      "11N36",
      "11R09",
      "11R11"
    ],
    "keywords": [
      "quadratic fields",
      "irreducible trinomials",
      "discriminant",
      "unconditional version",
      "character sums"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0811.1300S"
    }
  }
}