{
  "id": "2006.05417",
  "version": "v1",
  "published": "2020-06-09T17:31:01.000Z",
  "updated": "2020-06-09T17:31:01.000Z",
  "title": "On the distribution of the order and index for the reductions of algebraic numbers",
  "authors": [
    "Pietro Sgobba"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\alpha_1,\\ldots,\\alpha_r$ be algebraic numbers in a number field $K$ generating a torsion-free subgroup of rank $r$ in $K^\\times$. We investigate under GRH the number of primes $\\mathfrak p$ of $K$ such that each of the orders of $\\alpha_i\\bmod\\mathfrak p$ lies in a given arithmetic progression associated to $\\alpha_i$. We also study the primes $\\mathfrak p$ for which the index of $\\alpha_i\\bmod\\mathfrak p$ is a fixed integer or lies in a set of integers for each $i$. An additional condition on the Frobenius may be considered. Such results are generalizations of a theorem of Ziegler from 2006, which concerns the case $r=1$ of this problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-09T17:31:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R44",
      "11R45",
      "11R18",
      "11R21"
    ],
    "keywords": [
      "algebraic numbers",
      "distribution",
      "reductions",
      "number field",
      "additional condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}