{
  "id": "1202.5812",
  "version": "v1",
  "published": "2012-02-27T02:04:42.000Z",
  "updated": "2012-02-27T02:04:42.000Z",
  "title": "Noether's problem and unramified Brauer groups",
  "authors": [
    "Akinari Hoshi",
    "Ming-chang Kang",
    "Boris E. Kunyavskii"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:1109.2966",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $k$ be any field, $G$ be a finite group acing on the rational function field $k(x_g:g\\in G)$ by $h\\cdot x_g=x_{hg}$ for any $h,g\\in G$. Define $k(G)=k(x_g:g\\in G)^G$. Noether's problem asks whether $k(G)$ is rational (= purely transcendental) over $k$. It is known that, if $\\bm{C}(G)$ is rational over $\\bm{C}$, then $B_0(G)=0$ where $B_0(G)$ is the unramified Brauer group of $\\bm{C}(G)$ over $\\bm{C}$. Bogomolov showed that, if $G$ is a $p$-group of order $p^5$, then $B_0(G)=0$. This result was disproved by Moravec for $p=3,5,7$ by computer calculations. We will prove the following theorem. Theorem. Let $p$ be any odd prime number, $G$ be a group of order $p^5$. Then $B_0(G)\\ne 0$ if and only if $G$ belongs to the isoclinism family $\\Phi_{10}$ in R. James's classification of groups of order $p^5$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-02-27T02:04:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14E08",
      "14M20",
      "13A50",
      "20J06",
      "12F12"
    ],
    "keywords": [
      "unramified brauer group",
      "rational function field",
      "odd prime number",
      "noethers problem asks",
      "finite group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.5812H"
    }
  }
}