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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    W. Bäni and H. Gross, On SAP fields. Math. Z. 162 (1978) 69–74.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    C.J. Everett and H.J. Ryser, Rational vector spaces. Duke Mathematical Journal, vol. 16 (1949) 553–570.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    D. Hilbert, Grundlagen der Geometrie. Teubner, Stuttgart, 1956.zbMATHGoogle Scholar
  9. [9]
    S.S. Holland, Orderings and Square roots in *-fields. J. Alg. 46 (1977) 207–219.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    G. Maxwell, Classification of countably infinite hermitean forms over skewfields. Amer. J. Math. 96 (1974) 145–155.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    T. Szele, On ordered skew fields. Proc. Amer. Math. Soc. 3 (1952) 410–413.MathSciNetCrossRefzbMATHGoogle 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

Personalised recommendations