{
  "id": "2211.13822",
  "version": "v1",
  "published": "2022-11-24T23:16:18.000Z",
  "updated": "2022-11-24T23:16:18.000Z",
  "title": "Primes in denominators of algebraic numbers",
  "authors": [
    "Deepesh Singhal"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Denote the set of algebraic numbers as $\\overline{\\mathbb{Q}}$ and the set of algebraic integers as $\\overline{\\mathbb{Z}}$. For $\\gamma\\in\\overline{\\mathbb{Q}}$, consider its irreducible polynomial in $\\mathbb{Z}[x]$, $F_{\\gamma}(x)=a_nx^n+\\dots+a_0$. Denote $e(\\gamma)=\\gcd(a_{n},a_{n-1},\\dots,a_1)$. Drungilas, Dubickas and Jankauskas show in a recent paper that $\\mathbb{Z}[\\gamma]\\cap \\mathbb{Q}=\\{\\alpha\\in\\mathbb{Q}\\mid \\{p\\mid v_p(\\alpha)<0\\}\\subseteq \\{p\\mid p|e(\\gamma)\\}\\}$. Given a number field $K$ and $\\gamma\\in\\overline{\\mathbb{Q}}$, we show that there is a subset $X(K,\\gamma)\\subseteq \\text{Spec}(\\mathcal{O}_K)$, for which $\\mathcal{O}_K[\\gamma]\\cap K=\\{\\alpha\\in K\\mid \\{\\mathfrak{p}\\mid v_{\\mathfrak{p}}(\\alpha)<0\\}\\subseteq X(K,\\gamma)\\}$. We prove that $\\mathcal{O}_K[\\gamma]\\cap K$ is a principal ideal domain if and only if the primes in $X(K,\\gamma)$ generate the class group of $\\mathcal{O}_K$. We show that given $\\gamma\\in \\overline{\\mathbb{Q}}$, we can find a finite set $S\\subseteq \\overline{\\mathbb{Z}}$, such that for every number field $K$, we have $X(K,\\gamma)=\\{\\mathfrak{p}\\in\\text{Spec}(\\mathcal{O}_K)\\mid \\mathfrak{p}\\cap S\\neq \\emptyset\\}$. We study how this set $S$ relates to the ring $\\overline{\\mathbb{Z}}[\\gamma]$ and the ideal $\\mathfrak{D}_{\\gamma}=\\{a\\in\\overline{\\mathbb{Z}}\\mid a\\gamma\\in\\overline{\\mathbb{Z}}\\}$ of $\\overline{\\mathbb{Z}}$. We also show that $\\gamma_1,\\gamma_2\\in \\overline{\\mathbb{Q}}$ satisfy $\\mathfrak{D}_{\\gamma_1}=\\mathfrak{D}_{\\gamma_2}$ if and only if $X(K,\\gamma_1)=X(K,\\gamma_2)$ for all number fields $K$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-24T23:16:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "algebraic numbers",
      "number field",
      "denominators",
      "principal ideal domain",
      "algebraic integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}