Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References to Chapter XI
W. Bäni and H. Gross, On SAP fields. Math. Z. 162 (1978) 69–74.
R. Baer, Linear algebra and projective geometry. Academic Press New York, 1952.
N. Bourbaki, Algèbre chap. VI, groupes et corps ordonnés, ASI 1179, Hermann, Paris, 1952.
R. Elman, T.Y. Lam, A. Prestel, On some Hasse Principles over Formally Real Fields. Math. Z. 134 (1973) 291–301.
C. J. Everett and H.J. Ryser, Rational vector spaces. Duke Mathematical Journal, vol. 16 (1949) 553–570.
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.
H. Gross and H.R. Fischer, Non — real fields k and infinite dimensional k-vectorspaces. Mathematische Annalen, vol. 159 (1965) 285–308.
D. Hubert, Grundlagen der Geometrie. Teubner, Stuttgart, 1956.
S. S. Holland, Orderings and Square roots in *-fields. J. Alg. 46 (1977) 207–219.
I. Kaplansky, Forms in infinite — dimensional spaces, Anais da Academia Brasileira de Ciencias, vol. 22 (1950) 1–17.
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.
L. E. Mattics, Quadratic forms of countable dimension over algebraic number fields. Comment. Math. Helv. 43 (1968) 31–40.
G. Maxwell, Classification of countably infinite hermitean forms over skewfields. Amer. J. Math. 96 (1974) 145–155.
O. T. O’Meara, Infinite dimensional quadratic forms over algebraic number fields. Proc. Amer. Math. Soc. 10 (1959) 55–58.
A. Prestel, Quadratische Semi — Ordnungen und quadratische Formen. Math. Z. 133 (1973) 319–342.
A. Prestel and M. Ziegler, Erblich euklidische Körper. Journal reine angew. Math. 274/275 (1975) 196–205.
L. J. Savage, The application of vectorial methods to geometry. Duke Mathematical Journal, vol. 13 (1946) 521–528.
T. Szele, On ordered skew fields. Proc. Amer. Math. Soc. 3 (1952) 410–413.
E. Witt, Theorie der quadratischen Formen in beliebigen Körpern. J. reine angew. Math. 176 (1937) 31–44.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1979 Springer Science+Business Media New York
About this chapter
Cite this chapter
Gross, H. (1979). Classification of Forms Over Ordered Fields. In: Quadratic Forms in Infinite Dimensional Vector Spaces. Progress in Mathematics, vol 1. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-3542-7_12
Download citation
DOI: https://doi.org/10.1007/978-1-4899-3542-7_12
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-0-8176-1111-8
Online ISBN: 978-1-4899-3542-7
eBook Packages: Springer Book Archive