{
  "id": "1307.0743",
  "version": "v3",
  "published": "2013-07-02T16:08:50.000Z",
  "updated": "2014-10-22T17:42:30.000Z",
  "title": "First Order Decidability and Definability of Integers in Infinite Algebraic Extensions of Rational Numbers",
  "authors": [
    "Alexandra Shlapentokh"
  ],
  "comment": "Further revisions to improve readability: added a section with overview of the proof",
  "categories": [
    "math.NT",
    "math.LO"
  ],
  "abstract": "We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show the first-order undecidability. We also obtain a structural sufficient condition for definability of the ring of integers over its field of fractions. In particular, we show that the following propositions hold. (1) For any rational prime $q$ and any positive rational integer $m$, algebraic integers are definable in any Galois extension of Q where the degree of any finite subextension is not divisible by $q^{m}$. (2) Given a prime $q$, and an integer $m>0$, algebraic integers are definable in a cyclotomic extension (and any of its subfields) generated by any set $\\{\\xi_{p^{\\ell}}| \\ell \\in \\Z_{>0}, p \\not=q {is any prime such that} q^{m +1}\\not | (p-1)\\}$. (3) The first-order theory of any abelian extension of Q with finitely many ramified rational primes is undecidable. We also show that under a condition on the splitting of one rational prime in an infinite algebraic extension of Q, the existence of a finitely generated elliptic curve over the field in question is enough to have a definition of Z and to show that the field is indecidable.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-05-01T17:47:40.000Z",
      "abstract": "We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of $\\mathbb Q$ and use these definitions for some of the fields to show first-order undecidability. In particular, we show that the following propositions hold. (1) For any rational prime $q$ and any positive rational integer $m$, algebraic integers are definable in any Galois extension of $\\mathbb Q$ where the degree of any finite subextension is not divisible by $q^{m}$. (2) Given a prime $q$, and an integer $m>0$, algebraic integers are definable in a cyclotomic extension (and any of its subfields) generated by any set $\\{\\xi_{p^{\\ell}}| \\ell \\in \\mathbb Z_{>0}, p \\not=q \\mbox{ is any prime such that } q^{m +1}\\not | (p-1)\\}$. (3) The first-order theory of any abelian extension of $\\mathbb Q$ with finitely many ramified rational primes is undecidable.",
      "comment": "Major revisions to improve readability",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-10-22T17:42:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11U05"
    ],
    "keywords": [
      "infinite algebraic extensions",
      "first order decidability",
      "rational numbers",
      "definability",
      "rational prime"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.0743S"
    }
  }
}