{
  "id": "1803.01987",
  "version": "v1",
  "published": "2018-03-06T02:21:16.000Z",
  "updated": "2018-03-06T02:21:16.000Z",
  "title": "Odoni's conjecture for number fields",
  "authors": [
    "Robert L. Benedetto",
    "Jamie Juul"
  ],
  "comment": "15 pages, 1 figure",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Let $K$ be a number field, and let $d\\geq 2$. A conjecture of Odoni (stated more generally for characteristic zero Hilbertian fields $K$) posits that there is a monic polynomial $f\\in K[x]$ of degree $d$, and a point $x_0\\in K$, such that for every $n\\geq 0$, the so-called arboreal Galois group $Gal(K(f^{-n}(x_0))/K)$ is an $n$-fold wreath product of the symmetric group $S_d$. In this paper, we prove Odoni's conjecture when $d$ is even and $K$ is an arbitrary number field, and also when both $d$ and $[K:Q]$ are odd.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-06T02:21:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37P05",
      "11G50"
    ],
    "keywords": [
      "odonis conjecture",
      "characteristic zero hilbertian fields",
      "arbitrary number field",
      "arboreal galois group",
      "fold wreath product"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}