{
  "id": "1409.7829",
  "version": "v1",
  "published": "2014-09-27T18:53:07.000Z",
  "updated": "2014-09-27T18:53:07.000Z",
  "title": "Discriminants of simplest 3^n-tic extensions",
  "authors": [
    "T. Alden Gassert"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\ell>2$ be a positive integer, $\\zeta_\\ell$ a primitive $\\ell$-th root of unity, and $K$ a number field containing $\\zeta_\\ell+\\zeta_\\ell^{-1}$ but not $\\zeta_\\ell$. In a recent paper, Chonoles et. al. study iterated towers of number fields over $K$ generated by the generalized Rikuna polynomial, $r_n(x,t;\\ell) \\in K(t)[x]$. They note that when $K = \\mathbb{Q}$, $t \\in \\{0,1\\}$, and $\\ell=3$, the only ramified prime in the resulting tower is 3, and they ask under what conditions is the number of ramified primes small. In this paper, we apply a theorem of Gu\\`ardia, Montes, and Nart to derive a formula for the discriminant of $\\mathbb{Q}(\\theta)$ where $\\theta$ is a root of $r_n(x,t;3)$, answering the question of Chonoles et. al. in the case $K = \\mathbb{Q}$, $\\ell=3$, and $t \\in \\mathbb{Z}$. In the latter half of the paper, we identify some cases where the dynamics of $r_n(x,t;\\ell)$ over finite fields yields an explicit description of the decomposition of primes in these iterated extensions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-27T18:53:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "discriminant",
      "number field",
      "finite fields yields",
      "generalized rikuna polynomial",
      "study iterated towers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.7829G"
    }
  }
}