{
  "id": "math/0307055",
  "version": "v2",
  "published": "2003-07-03T19:47:29.000Z",
  "updated": "2005-06-24T20:07:01.000Z",
  "title": "The Beckman-Quarles theorem for mappings from R^2 to F^2, where F is a subfield of a commutative field extending R",
  "authors": [
    "Apoloniusz Tyszka"
  ],
  "comment": "LaTeX2e, 10 pages",
  "journal": "Abh. Math. Sem. Univ. Hamburg 74 (2004), 77-87",
  "categories": [
    "math.MG"
  ],
  "abstract": "Let F be a subfield of a commutative field extending R. Let \\phi_2: F^2 \\times F^2 \\to F, \\phi_2((x_1,x_2),(y_1,y_2))=(x_1-y_1)^2+(x_2-y_2)^2. We say that f:R^2 \\to F^2 preserves distance d \\geq 0 if for each x,y \\in R^2 |x-y|=d implies \\phi_2(f(x),f(y))=d^2. We prove that each unit-distance preserving mapping f:R^2 \\to F^2 has a form I \\circ (\\rho,\\rho), where \\rho: R \\to F is a field homomorphism and I: F^2 \\to F^2 is an affine mapping with orthogonal linear part.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-06-24T20:07:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51M05"
    ],
    "keywords": [
      "commutative field extending",
      "beckman-quarles theorem",
      "orthogonal linear part",
      "field homomorphism",
      "preserves distance"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......7055T"
    }
  }
}