{
  "id": "1312.6738",
  "version": "v2",
  "published": "2013-12-24T01:58:58.000Z",
  "updated": "2014-01-06T07:01:14.000Z",
  "title": "Class Numbers and Algebraic Tori",
  "authors": [
    "Akinari Hoshi",
    "Ming-chang Kang",
    "Aiichi Yamasaki"
  ],
  "comment": "A 2-page appendix is added",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $p$ be an odd prime number, $D_p$ be the dihedral group of order $2p$, $h_p$ and $h^+_p$ be the class numbers of $\\bm{Q}(\\zeta_p)$ and $\\bm{Q}(\\zeta_p+ \\zeta_p^{-1})$ respectively. Theorem. $h_p^+=1$ if and only if, for any field $k$ admitting a $D_p$-extension, all the algebraic $D_p$-tori over $k$ are stably rational. A similar result for $h_p=1$ and $C_p$-tori is valid also.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-01-06T07:01:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14E08",
      "11R33",
      "20C10"
    ],
    "keywords": [
      "class numbers",
      "algebraic tori",
      "odd prime number",
      "dihedral group",
      "similar result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.6738H"
    }
  }
}