{
  "id": "1810.00990",
  "version": "v1",
  "published": "2018-10-01T21:51:26.000Z",
  "updated": "2018-10-01T21:51:26.000Z",
  "title": "Finite index theorems for iterated Galois groups of unicritical polynomials",
  "authors": [
    "Andrew Bridy",
    "John R. Doyle",
    "Dragos Ghioca",
    "Liang-Chung Hsia",
    "Thomas J. Tucker"
  ],
  "comment": "20 pages",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Let $K$ be the function field of a smooth, irreducible curve defined over $\\overline{\\mathbb{Q}}$. Let $f\\in K[x]$ be of the form $f(x)=x^q+c$ where $q = p^{r}, r \\ge 1,$ is a power of the prime number $p$, and let $\\beta\\in \\overline{K}$. For all $n\\in\\mathbb{N}\\cup\\{\\infty\\}$, the Galois groups $G_n(\\beta)=\\mathop{\\rm{Gal}}(K(f^{-n}(\\beta))/K(\\beta))$ embed into $[C_q]^n$, the $n$-fold wreath product of the cyclic group $C_q$. We show that if $f$ is not isotrivial, then $[[C_q]^\\infty:G_\\infty(\\beta)]<\\infty$ unless $\\beta$ is postcritical or periodic. We are also able to prove that if $f_1(x)=x^q+c_1$ and $f_2(x)=x^q+c_2$ are two such distinct polynomials, then the fields $\\bigcup_{n=1}^\\infty K(f_1^{-n}(\\beta))$ and $\\bigcup_{n=1}^\\infty K(f_2^{-n}(\\beta))$ are disjoint over a finite extension of $K$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-01T21:51:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37P15",
      "11G50",
      "11R32",
      "14G25",
      "37P05",
      "37P30"
    ],
    "keywords": [
      "finite index theorems",
      "iterated galois groups",
      "unicritical polynomials",
      "fold wreath product",
      "prime number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}