{
  "id": "1905.06701",
  "version": "v1",
  "published": "2019-05-16T12:45:23.000Z",
  "updated": "2019-05-16T12:45:23.000Z",
  "title": "Fields of dimension one algebraic over a global or local field need not be of type $(C_{1})$",
  "authors": [
    "Ivan D. Chipchakov"
  ],
  "comment": "12 pages, LaTeX",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $(K, v)$ be a Henselian discrete valued field with a quasifinite residue field. This paper proves the existence of an algebraic extension $E/K$ with the following properties: (i) $E$ has dimension dim$(E) \\le 1$, i.e. the Brauer group Br$(E ^{\\prime })$ is trivial, for every algebraic extension $E ^{\\prime }/E$; (ii) no finite extension of $E$ is a $C _{1}$-field, in the sense of E. Artin and Lang. This, applied to the case where $K$ is the maximal algebraic extension of the field $\\mathbb Q$ of rational numbers in the field $\\mathbb Q _{p}$ of $p$-adic numbers, for a given prime number $p$, proves the existence of an algebraic extension $E _{p}$ of $\\mathbb Q$, such that dim$(E _{p}) \\le 1$, $E _{p}$ has a Henselian valuation with a residue field of characteristic $p$, and $E _{p}$ is not a $C _{1}$-field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-16T12:45:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E76",
      "11R34",
      "12J10",
      "11D72",
      "11S15"
    ],
    "keywords": [
      "local field",
      "henselian discrete valued field",
      "quasifinite residue field",
      "brauer group br",
      "maximal algebraic extension"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}