{
  "id": "1005.2016",
  "version": "v6",
  "published": "2010-05-12T08:40:08.000Z",
  "updated": "2011-10-31T06:39:09.000Z",
  "title": "Serre's \"formule de masse\" in prime degree",
  "authors": [
    "Chandan Singh Dalawat"
  ],
  "comment": "36 pages; most of the new material has been moved to the new Section 9",
  "journal": "Monatsh. Math. 166 (2012) 1, 73--92",
  "doi": "10.1007/s00605-010-0274-0",
  "categories": [
    "math.NT",
    "math.AC"
  ],
  "abstract": "For a local field F with finite residue field of characteristic p, we describe completely the structure of the filtered F_p[G]-module K^*/K^*p in characteristic 0 and $K^+/\\wp(K^+) in characteristic p, where K=F(\\root{p-1}\\of F^*) and G=\\Gal(K|F). As an application, we give an elementary proof of Serre's mass formula in degree p. We also determine the compositum C of all degree p separable extensions with solvable galoisian closure over an arbitrary base field, and show that C is K(\\root p\\of K^*) or K(\\wp^{-1}(K)) respectively, in the case of the local field F. Our method allows us to compute the contribution of each character G\\to\\F_p^* to the degree p mass formula, and, for any given group \\Gamma, the contribution of those degree p separable extensions of F whose galoisian closure has group \\Gamma.",
  "revisions": [
    {
      "version": "v6",
      "updated": "2011-10-31T06:39:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11S15",
      "11S20",
      "12F10"
    ],
    "keywords": [
      "prime degree",
      "local field",
      "galoisian closure",
      "separable extensions",
      "serres mass formula"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1005.2016S"
    }
  }
}