{
  "id": "1004.0986",
  "version": "v1",
  "published": "2010-04-06T23:48:39.000Z",
  "updated": "2010-04-06T23:48:39.000Z",
  "title": "Equidistribution of Algebraic Numbers of Norm One in Quadratic Number Fields",
  "authors": [
    "Kathleen L. Petersen",
    "Christopher D. Sinclair"
  ],
  "comment": "19 pages, 2 figures, comments welcome.",
  "categories": [
    "math.NT"
  ],
  "abstract": "Given a fixed quadratic extension K of Q, we consider the distribution of elements in K of norm 1 (denoted N). When K is an imaginary quadratic extension, N is naturally embedded in the unit circle in C and we show that it is equidistributed with respect to inclusion as ordered by the absolute Weil height. By Hilbert's Theorem 90, an element in N can be written as \\alpha/\\bar{\\alpha} for some \\alpha \\in O_K, which yields another ordering of \\mathcal N given by the minimal norm of the associated algebraic integers. When K is imaginary we also show that N is equidistributed in the unit circle under this norm ordering. When K is a real quadratic extension, we show that N is equidistributed with respect to norm, under the map \\beta \\mapsto \\log| \\beta | \\bmod{\\log | \\epsilon^2 |} where \\epsilon is a fundamental unit of O_K.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-04-06T23:48:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11K36"
    ],
    "keywords": [
      "quadratic number fields",
      "algebraic numbers",
      "equidistribution",
      "unit circle",
      "imaginary quadratic extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.0986P"
    }
  }
}