{
  "id": "1710.07357",
  "version": "v1",
  "published": "2017-10-19T21:20:25.000Z",
  "updated": "2017-10-19T21:20:25.000Z",
  "title": "Diophantine definability of nonnorms of cyclic extensions of global fields",
  "authors": [
    "Travis Morrison"
  ],
  "comment": "21 pages",
  "categories": [
    "math.NT",
    "math.LO"
  ],
  "abstract": "We show that for any square-free natural number $n$ and any global field $K$ with $(\\text{char}(K), n)=1$ containing the $n$th roots of unity, the pairs $(x,y)\\in K^*\\times K^*$ such that $x$ is not a norm of $K(\\sqrt[n]{y})/K$ form a diophantine set over $K$. We use the Hasse norm theorem, Kummer theory, and class field theory to prove this result. We also prove that for any $n\\in \\mathbb{N}$ and any global field $K$ with $\\text{char}(K)\\neq n$, $K^*\\setminus K^{*n}$ is diophantine over $K$. For a number field $K$, this is a result of Colliot-Th\\'el\\`ene and Van Geel, proved using results on the Brauer-Manin obstruction. Additionally, we prove a variation of our main theorem for global fields $K$ without the $n$th roots of unity, where we parametrize varieties arising from norm forms of cyclic extensions of $K$ without any rational points by a diophantine set.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-19T21:20:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "global field",
      "cyclic extensions",
      "diophantine definability",
      "th roots",
      "diophantine set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}