{
  "id": "1102.2043",
  "version": "v2",
  "published": "2011-02-10T06:26:26.000Z",
  "updated": "2011-04-14T18:21:41.000Z",
  "title": "Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis",
  "authors": [
    "Kevin J. McGown"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Assuming the Generalized Riemann Hypothesis (GRH), we show that the norm-Euclidean Galois cubic fields are exactly those with discriminant $\\Delta=7^2,9^2,13^2,19^2,31^2,37^2,43^2,61^2,67^2,103^2,109^2,127^2,157^2$. A large part of the proof is in establishing the following more general result: Let $K$ be a Galois number field of odd prime degree $\\ell$ and conductor $f$. Assume the GRH for $\\zeta_K(s)$. If $38(\\ell-1)^2(\\log f)^6\\log\\log f<f$, then $K$ is not norm-Euclidean.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-04-14T18:21:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A05",
      "11R04",
      "11R16",
      "11R80",
      "11Y40"
    ],
    "keywords": [
      "generalized riemann hypothesis",
      "norm-euclidean galois fields",
      "norm-euclidean galois cubic fields",
      "galois number field",
      "odd prime degree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1102.2043M"
    }
  }
}