{
  "id": "2009.02501",
  "version": "v1",
  "published": "2020-09-05T09:31:59.000Z",
  "updated": "2020-09-05T09:31:59.000Z",
  "title": "Galois groups of p-extensions of higher local fields modulo p-th commutators",
  "authors": [
    "Victor Abrashkin"
  ],
  "comment": "50 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Suppose $\\mathcal K$ is $N$-dimensional local field of characteristic $p$, $\\mathcal G =\\mathop{Gal}(\\mathcal K_{sep}/\\mathcal K)$, $\\mathcal G_{<p}$ is the maximal quotient of $\\mathcal G$ of period $p$ and nilpotent class $<p$ and $\\mathcal K_{<p}\\subset \\mathcal K_{sep}$ is such that $\\mathop{Gal}(\\mathcal K_{<p}/\\mathcal K)=\\mathcal G_{<p}$. We use nilpotent Artin-Schreier theory to identify $\\mathcal G_{<p}$ with the group $G(\\mathcal L)$ obtained from a profinite Lie $\\mathbb F_p$-algebra $\\mathcal L$ via the Campbell-Hausdorff composition law. The canonical $\\mathcal P$-topology on $\\mathcal K$ is used to define a dense Lie subalgebra $\\mathcal L^{\\mathcal P}$ in $\\mathcal L$. The algebra $\\mathcal L^{\\mathcal P}$ can be provided with a system of $\\mathcal P$-topological generators and its $\\mathcal P$-open subalgebras correspond to all $N$-dimensional extensions of $\\mathcal K$ in $\\mathcal K_{<p}$. These results are applied to higher local fields $K$ of characteristic 0 containing primitive $p$-th root of unity. If $\\Gamma =\\mathop{Gal}(K_{alg}/K)$ we introduce similarly the quotient $\\Gamma_{<p}=G(L)$, a dense $\\mathbb F_p$-Lie algebra $L^{\\mathcal P}\\subset L$, and describe the structure of $L^{\\mathcal P}$ in terms of generators and relations. The general result is illustrated by explicit presentation of $\\Gamma_{<p}$ modulo third commutators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-05T09:31:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11S15",
      "11S20"
    ],
    "keywords": [
      "local fields modulo p-th commutators",
      "higher local fields modulo p-th",
      "galois groups",
      "p-extensions",
      "dimensional local field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}