{
  "id": "0904.2416",
  "version": "v4",
  "published": "2009-04-16T10:55:01.000Z",
  "updated": "2011-03-03T03:17:24.000Z",
  "title": "On Brauer-Kuroda type relations of S-class numbers in dihedral extensions",
  "authors": [
    "Alex Bartel"
  ],
  "comment": "28 pages, restructured and expanded the proof of the main theorem, also some minor corrections. Final version, to appear in J. Reine Angew. Math",
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "Let F/k be a Galois extension of number fields with dihedral Galois group of order 2q, where q is an odd integer. We express a certain quotient of S-class numbers of intermediate fields, arising from Brauer-Kuroda relations, as a unit index. Our formula is valid for arbitrary extensions with Galois group D_{2q} and for arbitrary Galois-stable sets of primes S, containing the Archimedean ones. Our results have curious applications to determining the Galois module structure of the units modulo the roots of unity of a D_{2q}-extension from class numbers and S-class numbers. The techniques we use are mainly representation theoretic and we consider the representation theoretic results we obtain to be of independent interest.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2011-03-03T03:17:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R29",
      "11R33",
      "20C10",
      "11R27"
    ],
    "keywords": [
      "s-class numbers",
      "brauer-kuroda type relations",
      "dihedral extensions",
      "representation theoretic results",
      "dihedral galois group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0904.2416B"
    }
  }
}