{
  "id": "math/0310224",
  "version": "v2",
  "published": "2003-10-15T14:51:14.000Z",
  "updated": "2006-09-20T02:56:58.000Z",
  "title": "Integrality at a prime for global fields and the perfect closure of global fields of characteristic p>2",
  "authors": [
    "Kirsten Eisentraeger"
  ],
  "comment": "10 pages; minor revisions made",
  "journal": "J. Number Theory 114 (1) (2005), 170-181",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let k be a global field and \\pp any nonarchimedean prime of k. We give a new and uniform proof of the well known fact that the set of all elements of k which are integral at \\pp is diophantine over k. Let k^{perf} be the perfect closure of a global field of characteristic p>2. We also prove that the set of all elements of k^{perf} which are integral at some prime \\qq of k^{perf} is diophantine over k^{perf}, and this is the first such result for a field which is not finitely generated over its constant field. This is related to Hilbert's Tenth Problem because for global fields k of positive characteristic, giving a diophantine definition of the set of elements that are integral at a prime is one of two steps needed to prove that Hilbert's Tenth Problem for k is undecidable.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-09-20T02:56:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11U05",
      "03B25"
    ],
    "keywords": [
      "global field",
      "perfect closure",
      "characteristic",
      "integrality",
      "diophantine definition"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math.....10224E"
    }
  }
}