{
  "id": "1006.1156",
  "version": "v1",
  "published": "2010-06-07T01:54:58.000Z",
  "updated": "2010-06-07T01:54:58.000Z",
  "title": "The rationality problem for finite subgroups of GL_4(Q)",
  "authors": [
    "Ming-chang Kang",
    "Jian Zhou"
  ],
  "categories": [
    "math.AG",
    "math.AC",
    "math.RA"
  ],
  "abstract": "Let $G$ be a finite subgroup of $GL_4(\\bm{Q})$. The group $G$ induces an action on $\\bm{Q}(x_1,x_2,x_3,x_4)$, the rational function field of four variables over $\\bm{Q}$. Theorem. The fixed subfield $\\bm{Q}(x_1,x_2,x_3,x_4)^G:=\\{f\\in\\bm{Q}(x_1,x_2,x_3,x_4):\\sigma \\cdot f=f$ for any $\\sigma\\in G\\}$ is rational (i.e.\\ purely transcendental) over $\\bm{Q}$, except for two groups which are images of faithful representations of $C_8$ and $C_3\\rtimes C_8$ into $GL_4(\\bm{Q})$ (both fixed fields for these two exceptional cases are not rational over $\\bm{Q}$). There are precisely 227 such groups in $GL_4(\\bm{Q})$ up to conjugation; the answers to the rationality problem for most of them were proved by Kitayama and Yamasaki \\cite{KY} except for four cases. We solve these four cases left unsettled by Kitayama and Yamasaki; thus the whole problem is solved completely.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-06-07T01:54:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite subgroup",
      "rationality problem",
      "rational function field",
      "exceptional cases",
      "cases left"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1006.1156K"
    }
  }
}