{
  "id": "math/0506519",
  "version": "v4",
  "published": "2005-06-24T21:50:15.000Z",
  "updated": "2010-07-20T00:57:46.000Z",
  "title": "Geometric Galois Theory, Nonlinear Number Fields and a Galois Group Interpretation of the Idele Class Group",
  "authors": [
    "T. M. Gendron",
    "A. Verjovsky"
  ],
  "comment": "This is the revised version of an article which appeared by the same name in the International Journal of Mathematics, Vol. 16, No. 6 (July 2005)",
  "journal": "International Journal of Mathematics, Vol. 16, No. 6 (July 2005)",
  "categories": [
    "math.NT",
    "math.CV"
  ],
  "abstract": "This paper concerns the description of holomorphic extensions of algebraic number fields. We define a hyperbolized adele class group for every number field K Galois over Q and consider the Hardy space H[K] of graded-holomorphic functions on the hyperbolized adele class group. We show that the hyperplane N[K] in the projectivization PH[K] defined by the functions of non-zero trace possesses two partially-defined operations + and x, with respect to which there is canonical monomorphism of K into N[K]. We call N[K] a nonlinear field extension of K. We define Galois groups for nonlinear fields and show that Gal(N[L]/N[K]) is isomorphic to Gal(L/K) if L/K is Galois. If Q^{ab} denotes the maximal abelian extension of Q, C(Q) the idele class group and $\\bar{N}[Q^{ab}]=PH[K] is the full projectivization, then there are embeddings of C(Q) into Gal_{+}(\\bar{N}[Q^{ab}]/Q) and Gal_{x}(\\bar{N}[Q^{ab}]/Q), the \"Galois groups\" of automorphisms preserving + resp. x only.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2010-07-20T00:57:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R56",
      "11R37",
      "11R32",
      "57R30"
    ],
    "keywords": [
      "idele class group",
      "nonlinear number fields",
      "geometric galois theory",
      "galois group interpretation",
      "hyperbolized adele class group"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......6519G"
    }
  }
}