{
  "id": "1403.0318",
  "version": "v2",
  "published": "2014-03-03T05:59:37.000Z",
  "updated": "2014-03-07T02:38:45.000Z",
  "title": "Noether's problems for groups of order 243",
  "authors": [
    "Huah Chu",
    "Akinari Hoshi",
    "Shou-Jen Hu",
    "Ming-chang Kang"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1201.5555 by other authors",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $k$ be any field, $G$ be a finite group. Let $G$ act on the rational function field $k(x_g:g\\in G)$ by $k$-automorphisms defined by $h\\cdot x_g=x_{hg}$ for any $g,h\\in G$. Denote by $k(G)=k(x_g:g\\in G)^G$ the fixed field. Noether's problem asks, under what situations, the fixed field $k(G)$ will be rational (= purely transcendental) over $k$. According to the data base of GAP there are $10$ isoclinism families for groups of order $243$. It is known that there are precisely $3$ groups $G$ of order $243$ (they consist of the isoclinism family $\\Phi_{10}$) such that the unramified Brauer group of $\\bm{C}(G)$ over $\\bm{C}$ is non-trivial. Thus $\\bm{C}(G)$ is not rational over $\\bm{C}$. We will prove that, if $\\zeta_9 \\in k$, then $k(G)$ is rational over $k$ for groups of order $243$ other than these $3$ groups, except possibly for groups belonging to the isoclinism family $\\Phi_7$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-03-07T02:38:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13A50"
    ],
    "keywords": [
      "isoclinism family",
      "fixed field",
      "rational function field",
      "noethers problem asks",
      "unramified brauer group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.0318C"
    }
  }
}