Classification of Forms over Ordered Fields

  • Herbert Gross
Part of the Progress in Mathematics book series (PM, volume 1)


In this chapter we shall show that a certain kind of commutative ordered fields, the so called SAP fields, lend themselves very naturally for the construction of ℵo-forms which admit a simple classification with respect to isometry. We shall first say a few words about the fields and then describe the type of ℵo-forms to be studied.


Stable Form Hyperbolic Plane Isotropic Vector Algebraic Number Field Canonical Representative 


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  1. [1]
    W. Bäni and H. Gross, On SAP fields. Math. Z. 162 (1978) 69–74.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    R. Baer, Linear algebra and projective geometry. Academic Press New York, 1952.Google Scholar
  3. [3]
    N. Bourbaki, Algèbre chap. VI, groupes et corps ordonnés,ASI 1179, Hermann, Paris, 1952.Google Scholar
  4. [4]
    R. Elman, T.Y. Lam, A. Prestel, On some Hasse Principles over Formally Real Fields. Math. Z. 134 (1973) 291–301.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    C.J. Everett and H.J. Ryser, Rational vector spaces. Duke Mathematical Journal, vol. 16 (1949) 553–570.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    H. Gross and R.D. Engle, Bilinear forms on k-vectorspaces of denumerable dimension in the case of char (k) =2, Commentarii Mathematici Helvetici, vol. 40 (1965) 247–266.MathSciNetCrossRefGoogle Scholar
  7. [7]
    H. Gross and H.R. Fischer, Non-real fields k and infinite di-mensional k-vectorspaces. Mathematische Annalen, vol. 159 (1965) 285–308.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    D. Hilbert, Grundlagen der Geometrie. Teubner, Stuttgart, 1956.MATHGoogle Scholar
  9. [9]
    S.S. Holland, Orderings and Square roots in *-fields. J. Alg. 46 (1977) 207–219.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    I. Kaplansky, Forms in infinite-dimensional spaces, Anais da Academia Brasileira de Ciencias, vol. 22 (1950) 1–17.MathSciNetGoogle Scholar
  11. [11]
    M. Knebusch, A. Rosenberg, R. Ware, Structure of Witt rings, quotients of abelian group rings, and orderings of fields. Bull. Amer. Math. Soc. 77 (1971) 205–210.Google Scholar
  12. [12]
    L.E. Mattics, Quadratic forms of countable dimension over algebraic number fields. Comment. Math. Helv. 43 (1968) 31–40.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    G. Maxwell, Classification of countably infinite hermitean forms over skewfields. Amer. J. Math. 96 (1974) 145–155.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    O.T. O’Meara, Infinite dimensional quadratic forms over algebraic number fields. Proc. Amer. Math. Soc. 10 (1959) 55–58.Google Scholar
  15. [15]
    A. Prestel, Quadratische Semi–Ordnungen und quadratische Formen. Math. Z. 133 (1973) 319–342.Google Scholar
  16. [16]
    A. Prestel and M. Ziegler, Erblich euklidische Körper. Journal reine angew. Math. 274 /275 (1975) 196–205.MathSciNetGoogle Scholar
  17. [17]
    L.J. Savage, The application of vectorial methods to geometry. Duke Mathematical Journal, vol. 13 (1946) 521–528.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    T. Szele, On ordered skew fields. Proc. Amer. Math. Soc. 3 (1952) 410–413.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    E. Witt, Theorie der quadratischen Formen in beliebigen Körpern. J. reine angew. Math. 176 (1937) 31–44Google Scholar

Copyright information

© Springer Science+Business Media New York 1979

Authors and Affiliations

  • Herbert Gross
    • 1
  1. 1.Mathematisches InstitutUniversität ZürichZürichSwitzerland

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