{
  "id": "1903.03750",
  "version": "v1",
  "published": "2019-03-09T06:56:28.000Z",
  "updated": "2019-03-09T06:56:28.000Z",
  "title": "An application of cohomological invariants",
  "authors": [
    "Akinari Hoshi",
    "Ming-chang Kang",
    "Aiichi Yamasaki"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $G$ be a finite group, $k$ be a field and $G\\to GL(V_{\\rm reg})$ be the regular representation of $G$ over $k$. Then $G$ acts naturally on the rational function field $k(V_{\\rm reg})$ by $k$-automorphisms. Define $k(G)$ to be the fixed field $k(V_{\\rm reg})^G$. Noether's problem asks whether $k(G)$ is rational (resp. stably rational) over $k$. When $k=\\bQ$ and $G$ contains a normal subgroup $N$ with $G/H\\simeq C_8$ (the cyclic group of order $8$), Jack Sonn proves that $\\bQ(G)$ is not stably rational over $\\bQ$, which is a non-abelian extension of a theorem of Endo-Miyata, Voskresenskii, Lenstra and Saltman for the abelian Noether's problem $\\bQ(C_8)$. Using the method of cohomological invariants, we are able to generalize Sonn's theorem as follows. Theorem. Let $G$ be a finite group and $N$ $\\lhd$ $G$ such that $G/N\\simeq C_{2^n}$ with $n\\geq 3$. If $k$ is a field satisfying that ${\\rm char}\\,k=0$ and $k(\\zeta_{2^n})/k$ is not a cyclic extension where $\\zeta_{2^n}$ is a primitive $2^n$-th root of unity, then $k(G)$ is not stably rational (resp. not retract rational) over $k$. \\end{abstract}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-09T06:56:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "12G05",
      "14E08"
    ],
    "keywords": [
      "cohomological invariants",
      "stably rational",
      "application",
      "finite group",
      "abelian noethers problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}