{
  "id": "1608.07350",
  "version": "v1",
  "published": "2016-08-26T02:13:58.000Z",
  "updated": "2016-08-26T02:13:58.000Z",
  "title": "Extensions of local fields and elementary symmetric polynomials",
  "authors": [
    "Kevin Keating"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $K$ be a local field whose residue field has characteristic $p$ and let $L/K$ be a finite separable totally ramified extension of degree $n=up^{\\nu}$. Let $\\sigma_1,\\dots,\\sigma_n$ denote the $K$-embeddings of $L$ into a separable closure $K^{sep}$ of $K$. For $1\\le h\\le n$ let $e_h(X_1,\\dots,X_n)$ denote the $h$th elementary symmetric polynomial in $n$ variables, and for $\\alpha\\in L$ set $E_h(\\alpha) =e_h(\\sigma_1(\\alpha),\\dots,\\sigma_n(\\alpha))$. Set $j=\\min\\{v_p(h),\\nu\\}$. We show that for $r\\in\\mathbb{Z}$ we have $E_h(\\mathcal{M}_L^r)\\subset \\mathcal{M}_K^{\\lceil(i_j+hr)/n\\rceil}$, where $i_j$ is the $j$th index of inseparability of $L/K$. In certain cases we also show that $E_h(\\mathcal{M}_L^r)$ is not contained in any higher power of $\\mathcal{M}_K$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-26T02:13:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11S15"
    ],
    "keywords": [
      "local field",
      "th elementary symmetric polynomial",
      "higher power",
      "residue field",
      "finite separable totally ramified extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}