{
  "id": "1511.06446",
  "version": "v1",
  "published": "2015-11-19T23:27:46.000Z",
  "updated": "2015-11-19T23:27:46.000Z",
  "title": "Galois Groups of Generic Polynomials",
  "authors": [
    "Igor Rivin"
  ],
  "comment": "12pp",
  "categories": [
    "math.NT"
  ],
  "abstract": "We show that the Galois group of a random monic polynomial %of degree $d>12$ with integer coefficients between $-N$ and $N$ is NOT $S_d$ with probability $\\ll \\frac{\\log^{\\Omega(d)}N}{N}.$ Conditionally on NOTbeing the full symmetric group, we have a hierarchy of possibilities each of which has polylog probability of occurring. These results also apply to random polynomials with only a subset of the coefficients allowed to vary. This settles a question going back to 1936.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-19T23:27:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R45",
      "11C08",
      "11R32"
    ],
    "keywords": [
      "galois group",
      "generic polynomials",
      "random monic polynomial",
      "full symmetric group",
      "random polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151106446R"
    }
  }
}