{
  "id": "1704.03124",
  "version": "v1",
  "published": "2017-04-11T03:11:59.000Z",
  "updated": "2017-04-11T03:11:59.000Z",
  "title": "Counting $G$-Extensions by Discriminant",
  "authors": [
    "Evan P. Dummit"
  ],
  "comment": "14 pages. Comments welcome!",
  "categories": [
    "math.NT"
  ],
  "abstract": "The problem of analyzing the number of number field extensions $L/K$ with bounded (relative) discriminant has been the subject of renewed interest in recent years, with significant advances made by Schmidt, Ellenberg-Venkatesh, Bhargava, Bhargava-Shankar, and others. In this paper, we use the geometry of numbers and invariant theory of finite groups, in a manner similar to Ellenberg and Venkatesh, to give an upper bound on the number of extensions $L/K$ with fixed degree, bounded relative discriminant, and specified Galois closure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-11T03:11:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R21",
      "11R29",
      "13A50",
      "11H06"
    ],
    "keywords": [
      "discriminant",
      "number field extensions",
      "invariant theory",
      "specified galois closure",
      "upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}