{
  "id": "1611.06783",
  "version": "v1",
  "published": "2016-11-21T13:45:58.000Z",
  "updated": "2016-11-21T13:45:58.000Z",
  "title": "Cyclotomic polynomials at roots of unity",
  "authors": [
    "Bartlomiej Bzdega",
    "Andres Herrera-Poyatos",
    "Pieter Moree"
  ],
  "comment": "17 pages, 3 tables",
  "categories": [
    "math.NT"
  ],
  "abstract": "The $n^{th}$ cyclotomic polynomial $\\Phi_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Hence $\\Phi_n(x)$ is trivially zero at $n^{th}$ roots of unity. We investigate the values of $\\Phi_n(x)$ at the other roots of unity. Motose (2006) seems to have been the first to evaluate $\\Phi_n(e^{2\\pi i/m})$ with $m\\in \\{3,4,6\\}$. We reprove and correct his work. Furthermore, we give a simple reproof of a result of Apostol (1970) on the resultant of cyclotomic polynomials and of a result of Vaughan (1975) on the maximum coefficient (in absolute value) of $\\Phi_n(x)$. Also we precisely characterize the $n$ and $m$ for which $\\Phi_n(\\zeta_m)\\in \\{\\pm 1\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-21T13:45:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cyclotomic polynomial",
      "minimal polynomial",
      "maximum coefficient",
      "simple reproof",
      "absolute value"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}