{
  "id": "math/0606368",
  "version": "v1",
  "published": "2006-06-15T15:38:40.000Z",
  "updated": "2006-06-15T15:38:40.000Z",
  "title": "Diophantine Definability and Decidability in the Extensions of Degree 2 of Totally Real Fields",
  "authors": [
    "Alexandra Shlapentokh"
  ],
  "categories": [
    "math.NT",
    "math.LO"
  ],
  "abstract": "We investigate Diophantine definability and decidability over some subrings of algebraic numbers contained in quadratic extensions of totally real algebraic extensions of $\\mathbb Q$. Among other results we prove the following. The big subring definability and undecidability results previously shown by the author to hold over totally complex extensions of degree 2 of totally real number fields, are shown to hold for {\\it all} extensions of degree 2 of totally real number fields. The definability and undecidability results for integral closures of ``small'' and ``big'' subrings of number fields in the infinite algebraic extensions of $\\mathbb Q$, previously shown by the author to hold for totally real fields, are extended to a large class of extensions of degree 2 of totally real fields. This class includes infinite cyclotomics and abelian extensions with finitely many ramified rational primes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-06-15T15:38:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11U05",
      "03D35"
    ],
    "keywords": [
      "totally real fields",
      "diophantine definability",
      "totally real number fields",
      "undecidability results",
      "infinite algebraic extensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......6368S"
    }
  }
}