{
  "id": "math/0502565",
  "version": "v35",
  "published": "2005-02-27T07:05:14.000Z",
  "updated": "2006-04-18T15:40:25.000Z",
  "title": "On \\emptyset-definable elements in a field",
  "authors": [
    "Apoloniusz Tyszka"
  ],
  "comment": "15 pages, LaTeX2e, the version which will appear in Collectanea Mathematica",
  "journal": "Collectanea Mathematica 58 (2007), no. 1, pp. 73-84",
  "categories": [
    "math.LO",
    "math.NT"
  ],
  "abstract": "Let K be a field and \\tilde{K} denote the set of all r \\in K for which there exists a finite set A(r) with {r} \\subseteq A(r) \\subseteq K such that each mapping f:A(r) \\to K that satisfies: if 1 \\in A(r) then f(1)=1, if a,b \\in A(r) and a+b \\in A(r) then f(a+b)=f(a)+f(b), if a,b \\in A(r) and a \\cdot b \\in A(r) then f(a \\cdot b)=f(a) \\cdot f(b), satisfies also f(r)=r. We prove: \\tilde{K} is a subfield of K, \\tilde{K}={x \\in K: {x} is existentially first-order definable in the language of rings without parameters}, if some subfield of K is algebraically closed then \\tilde{K} is the prime field in K, some elements of \\tilde{K} are transcendental over Q (over R, over Q_p) for a large class of fields K that are finitely generated over Q (that extend R, that extend Q_p), if K is a Pythagorean subfield of R, t is transcendental over K, and r \\in K is recursively approximable, then {r} is \\emptyset-definable in (K(t),+,\\cdot,0,1), if a real number r is recursively approximable then {r} is existentially \\emptyset-definable in (R,+,\\cdot,0,1,U) for some unary predicate U which is implicitly \\emptyset-definable in (R,+,\\cdot,0,1).",
  "revisions": [
    {
      "version": "v35",
      "updated": "2006-04-18T15:40:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C60",
      "12L12"
    ],
    "keywords": [
      "real number",
      "finite set",
      "prime field",
      "pythagorean subfield",
      "large class"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......2565T"
    }
  }
}