{
  "id": "math/0609723",
  "version": "v5",
  "published": "2006-09-26T10:01:01.000Z",
  "updated": "2007-01-10T09:30:03.000Z",
  "title": "On prime factors of class number of cyclotomic fields",
  "authors": [
    "Roland Queme"
  ],
  "comment": "The numerical MAPLE algorithm of the section 5 p. 18 is added",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let p be an odd prime. Let K = \\Q(zeta) be the p-cyclotomic number field. Let v be a primitive root mod p and sigma : zeta --> zeta^v be a \\Q-isomorphism of the extension K/\\Q generating the Galois group G of K/\\Q. For n in Z, the notation v^n is understood by v^n mod p with 1 \\leq v^n \\leq p-1. Let P(X) = \\sum_{i=0}^{p-2} v^{-i}X^i \\in \\Z[X] be the Stickelberger polynomial. P(sigma) annihilates the class group C of K. There exists a polynomial Q(X) \\in \\Z[X] such that P(sigma)(sigma-v) = p\\times Q(sigma) and such that Q(sigma) annihilates the p-class group C_p of K (the subgroup of exponent p of C). In the other hand sigma^{(p-1)/2}+1 annihilates the relative class group of K. The simultaneous application of these results brings some informations on the structure of the class group C, give some explicit congruences in \\Z[v] mod p for the p-class group C_p of K and some explicit congruences in \\Z[v] mod h for the h-class group of K for all the prime divisors h \\not = p of the class number h(K). We detail at the end the case of class number of quadratic and biquadratic fields contained in the cyclotomic field K and give a general MAPLE algorithm.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2007-01-10T09:30:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R18",
      "11R29"
    ],
    "keywords": [
      "class number",
      "cyclotomic field",
      "prime factors",
      "explicit congruences",
      "p-class group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......9723Q"
    }
  }
}