{
  "id": "2011.11135",
  "version": "v1",
  "published": "2020-11-22T23:25:26.000Z",
  "updated": "2020-11-22T23:25:26.000Z",
  "title": "On Fields of dimension one that are Galois extensions of a global or local field",
  "authors": [
    "Ivan D. Chipchakov"
  ],
  "comment": "LaTeX, 18 pages, no figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $K$ be a global or local field, $E/K$ a Galois extension, and Br$(E)$ the Brauer group of $E$. This paper shows that if $K$ is a local field, $v$ is its natural discrete valuation, $v'$ is the valuation of $E$ extending $v$, and $q$ is the characteristic of the residue field $\\widehat E$ of $(E, v')$, then Br$(E) = \\{0\\}$ if and only if the following conditions hold: $\\widehat E$ contains as a subfield the maximal $p$-extension of $\\widehat K$, for each prime $p \\neq q$; $\\widehat E$ is an algebraically closed field in case the value group $v'(E)$ is $q$-indivisible. When $K$ is a global field, it characterizes the fields $E$ with Br$(E) = \\{0\\}$, which lie in the class of tame abelian extensions of $K$. We also give a criterion that, in the latter case, for any integer $n \\ge 2$, there exists an $n$-variate $E$-form of degree $n$, which violates the Hasse principle.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-22T23:25:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E76",
      "11R34",
      "12J10",
      "11D72",
      "11S15"
    ],
    "keywords": [
      "local field",
      "galois extension",
      "natural discrete valuation",
      "tame abelian extensions",
      "hasse principle"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}