{
  "id": "2303.02521",
  "version": "v2",
  "published": "2023-03-04T22:53:37.000Z",
  "updated": "2023-05-07T21:11:52.000Z",
  "title": "Defining $\\mathbb Z$ using unit groups",
  "authors": [
    "Barry Mazur",
    "Karl Rubin",
    "Alexandra Shlapentokh"
  ],
  "comment": "Minor changes and corrections",
  "categories": [
    "math.NT",
    "math.LO"
  ],
  "abstract": "We consider first-order definability and decidability questions over rings of integers of algebraic extensions of $\\mathbb Q$, paying attention to the uniformity of definitions. The uniformity follows from the simplicity of our first-order definition of $\\mathbb Z$. Namely, we prove that for a large collection of algebraic extensions $K/\\mathbb Q$, $$ \\{x \\in {\\mathcal O}_K : \\text{$\\forall \\varepsilon \\in {\\mathcal O}_K^\\times \\;\\exists \\delta \\in {\\mathcal O}_K^\\times$ such that $\\delta-1 \\equiv (\\varepsilon-1)x \\pmod{(\\varepsilon-1)^2}$}\\} = \\mathbb Z $$ where ${\\mathcal O}_K$ denotes the ring of integers of $K$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-05-07T21:11:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11U05"
    ],
    "keywords": [
      "unit groups",
      "algebraic extensions",
      "first-order definability",
      "large collection",
      "decidability questions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}