{
  "id": "1107.4595",
  "version": "v4",
  "published": "2011-07-22T19:04:24.000Z",
  "updated": "2014-05-19T08:36:45.000Z",
  "title": "The order of the reductions of an algebraic integer",
  "authors": [
    "Antonella Perucca"
  ],
  "comment": "revised and rewritten; new tables of examples checked with sage; the revisions include the referee's comments",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let K be a number field, and let a be a non-zero element of K. Fix some prime number l. We compute the density of the following set: the primes p of K such that the multiplicative order of the reduction of a modulo p is coprime to l (or, more generally, has some prescribed l-adic valuation). We evaluate the degree over K of extensions of the form K(\\zeta_m, \\sqrt[n]{a}) with n\\leq m, which are obtained by adjoining a root of unity of order l^m and the l^n-th roots of a, as this is needed for computing the above density.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-05-19T08:36:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R44",
      "11R18",
      "11Y40"
    ],
    "keywords": [
      "algebraic integer",
      "non-zero element",
      "number field",
      "prescribed l-adic valuation",
      "prime number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1107.4595P"
    }
  }
}