{
  "id": "math/0110055",
  "version": "v1",
  "published": "2001-10-04T15:50:08.000Z",
  "updated": "2001-10-04T15:50:08.000Z",
  "title": "Enumeration of Isomorphism Classes of Extensions of p-adic Fields",
  "authors": [
    "Xiang-dong Hou",
    "Kevin Keating"
  ],
  "comment": "39 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $\\Omega$ be an algebraic closure of ${\\mathbb Q}_p$ and let $F$ be a finite extension of ${\\mathbb Q}_p$ contained in $\\Omega$. Given positive integers $f$ and $e$, the number of extensions $K/F$ contained in $\\Omega$ with residue degree $f$ and ramification index $e$ was computed by Krasner. This paper is concerned with the number ${\\mathfrak I}(F,f,e)$ of $F$-isomorphism classes of such extensions. We determine ${\\mathfrak I}(F,f,e)$ completely when $p^2\\nmid e$ and get partial results when $p^2\\parallel e$. When $s$ is large, ${\\mathfrak I}({\\mathbb Q}_p,f,e)$ is equal to the number of isomorphism classes of finite commutative chain rings with residue field ${\\mathbb F}_{p^f}$, ramification index $e$, and length $s$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-10-04T15:50:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11S15"
    ],
    "keywords": [
      "isomorphism classes",
      "p-adic fields",
      "ramification index",
      "enumeration",
      "finite commutative chain rings"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math.....10055H"
    }
  }
}