Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-12-04T10:14:48.657Z Has data issue: false hasContentIssue false
  • English
  • Français

A Witt Theorem for Non-Defective Lattices

Published online by Cambridge University Press:  20 November 2018

Karl A. Morin-Strom
Karl A. Morin-Strom*
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts; McGill University, Montreal, Quebec
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In [10], Witt laid the foundation for the study of quadratic forms over fields. Suppose Q is a quadratic form defined on a finite dimensional vector space V over a field of characteristic not equal to 2. Witt showed that non-zero vectors x and y in V satisfying Q(x) = Q(y) can be mapped into each other via an isometry of the vector space V. More generally, if τ : W ⟶ W’ is an isometry between subspaces of V, then τ extends to an isometry ϕ of V.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 3 , 01 June 1978 , pp. 499 - 511
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Band, M., On the integral extensions of quadratic forms over local fields, Can. J. Math. 22 (1970), 297–307.Google Scholar
2. Cohen, D. M., Witt's theorem for quadratic forms, Conference on Quadratic Forms, Queen's Papers in Mathematics, 46 (1977), 406–411.Google Scholar
3. Hsia, J. S., note on the integral equivalence of vectors in characteristic 2, Math. Ann. 179 (1968), 63–69.Google Scholar
4. Hsia, J. S., One dimensional Witt's theorem over modular lattices, Bull. Amer. Math. Soc. 76 (1970), 113–115.Google Scholar
5. James, D. G. and Rosenzweig, S. M., Associated vectors in lattices over valuation rings, Amer. J. Math. 90 (1968), 295–307.Google Scholar
6. Kneser, M., Witts Satz fur quadratische Formen uber lokalen Ringen, Nachr. die Akad. der Wiss. Gottingen, Math.-Phys. II Heft 9 (1972), 195–203.Google Scholar
7. O'Meara, O. T., Introduction to quadratic forms, Grundlchren der Math. Wiss. (Springer-Verlag, Berlin 1971).Google Scholar
8. Rosenzweig, S. M., An anology of Witt's theorem for modules over the ring of p-adic integers, Ph.D. thesis, M.I.T. (1958).Google Scholar
9. Trojan, A., The integral extension of isometrics of quadratic forms over local fields, Can. J. Math. 18 (1966), 920–942.Google Scholar
10. Witt, E., Théorie der quadratischen Formen in beliebigen Korpen, Journal fur die reine und angewandte Math. 176 (1937), 31–44.Google Scholar