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A Witt Theorem for Non-Defective Lattices

  • Karl A. Morin-Strom (a1)

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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 : WW’ is an isometry between subspaces of V, then τ extends to an isometry ϕ of V.

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References

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1. Band, M., On the integral extensions of quadratic forms over local fields, Can. J. Math. 22 (1970), 297307.
2. Cohen, D. M., Witt's theorem for quadratic forms, Conference on Quadratic Forms, Queen's Papers in Mathematics, 46 (1977), 406411.
3. Hsia, J. S., note on the integral equivalence of vectors in characteristic 2, Math. Ann. 179 (1968), 6369.
4. Hsia, J. S., One dimensional Witt's theorem over modular lattices, Bull. Amer. Math. Soc. 76 (1970), 113115.
5. James, D. G. and Rosenzweig, S. M., Associated vectors in lattices over valuation rings, Amer. J. Math. 90 (1968), 295307.
6. Kneser, M., Witts Satz fur quadratische Formen uber lokalen Ringen, Nachr. die Akad. der Wiss. Gottingen, Math.-Phys. II Heft 9 (1972), 195203.
7. O'Meara, O. T., Introduction to quadratic forms, Grundlchren der Math. Wiss. (Springer-Verlag, Berlin 1971).
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).
9. Trojan, A., The integral extension of isometrics of quadratic forms over local fields, Can. J. Math. 18 (1966), 920942.
10. Witt, E., Théorie der quadratischen Formen in beliebigen Korpen, Journal fur die reine und angewandte Math. 176 (1937), 3144.
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A Witt Theorem for Non-Defective Lattices

  • Karl A. Morin-Strom (a1)

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