{
  "id": "1810.09710",
  "version": "v1",
  "published": "2018-10-23T08:14:22.000Z",
  "updated": "2018-10-23T08:14:22.000Z",
  "title": "Number systems over orders of finite étale algebras",
  "authors": [
    "Kálmán Győry",
    "Attila~Pethő",
    "Jörg~Thuswaldner"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\K$ be an algebraic number field and $\\Omega$ a finite \\'etale $\\K$-algebra. Denote by $\\Z_{\\Omega}$ the ring of integers of $\\Omega$. Generalizing the recently introduced number systems over orders of number fields we introduce in this paper so-called \\emph{\\'etale number systems} over $\\Z_{\\Omega}$ (ENS for short). An ENS is a pair $(p,\\DD)$, where $p\\in \\Z_{\\Omega}[x]$ is monic and $\\DD$ is a complete residue system modulo $p(0)$ in $\\Z_{\\Omega}$. ENS with finiteness property, {\\it i.e.}, with the property that all elements of $\\Z_{\\Omega}/(p)$ have a representative belonging to $\\DD[x],$ play an important role. We prove that it is algorithmically decidable whether or not an ENS admits the finiteness property. Under mild conditions we show that the pairs $(p(x+\\alpha),\\DD), p\\in \\Z_{\\Omega}[x]$ are always ENS with finiteness property provided $\\alpha\\in \\Z_{\\Omega}$ is in some sense large enough, for example, if $\\alpha$ is a sufficiently large rational integer. In the opposite direction we prove under different conditions that $(p(x-m),\\DD)$ does not have the finiteness property for each large enough rational integer $m$. We obtain important relations between power integral bases and ENS in orders of \\'etale $\\Q$-algebras. The proofs depend on some general effective finiteness results of Evertse and Gy\\H{o}ry on monogenic orders in \\'etale algebras. The paper ends with some speculations on possible further generalizations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-23T08:14:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A63",
      "11R04"
    ],
    "keywords": [
      "number systems",
      "finiteness property",
      "complete residue system modulo",
      "general effective finiteness results",
      "power integral bases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}