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Decomposition of Witt Rings

Published online by Cambridge University Press:  20 November 2018

Andrew B. Carson  and
Murray A. Marshall
Andrew B. Carson
Affiliation:
University of Saskatchewan, Saskatoon, Saskatchewan
Murray A. Marshall
Affiliation:
University of Saskatchewan, Saskatoon, Saskatchewan
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We take the definition of a Witt ring to be that given in [13], i.e., it is what is called a strongly representational Witt ring in [8]. The classical example is obtained by considering quadratic forms over a field of characteristic ≠ 2 [17], but Witt rings also arise in studying quadratic forms or symmetric bilinear forms over more general types of rings [5,7, 8, 9]. An interesting problem in the theory is that of classifying Witt rings in case the associated group G is finite. The reduced case, i.e., the case where the nilradical is trivial, is better understood. In particular, the above classification problem is completely solved in this case [4, 12, or 13, Corollary 6.25]. Thus, the emphasis here is on the non-reduced case. Although some of the results given here do not require |G| < ∞, they do require some finiteness assumption. Certainly, the main goal here is to understand the finite case, and in this sense this paper is a continuation of work started by the second author in [13, Chapter 5].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 6 , 01 December 1982 , pp. 1276 - 1302
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Berman, L., Cordes, C. and Ware, R., Quadratic forms, rigid elements, and formal power series fields, J. of Alg. 66 (1980), 123–133.Google Scholar
2. Cordes, C., Quadratic forms over non-formally real fields with a finite number of quaternion algebras, Pac. J. Math. 63 (1976), 357–365.Google Scholar
3. Cordes, C. and Ramsey, J. Jr., Quadratic forms over fields with u = q/2 < + °°, Fund. Math. 99 (1978), 1–10.Google Scholar
4. Craven, T., Characterizing reduced Witt rings of fields, J. of Alg. 53 (1978), 68–77.Google Scholar
5. Craven, T., Witt rings and orderings of skew fields, preprint.Google Scholar
6. Harrison, D., Witt rings, notes by J. Cunningham, Univ. of Kentucky (1970).Google Scholar
7. Kirkwood, B. and McDonald, B., in preparation.Google Scholar
8. Kleinstein, J. and Rosenberg, A., Succinct and representational Witt rings, Pac. J. Math. 86 (1980), 99–137.Google Scholar
9. Knebusch, M., On the local theory of signatures and reduced quadratic forms, preprint.Google Scholar
10. Kula, M., Fields with prescribed quadratic form schemes, Math. Zeit. 167 (1979), 201–212.Google Scholar
11. Kula, M., Szczepanik, L. and Szymiczek, K., Quadratic forms over formally real fields with eight square classes, Manuscripta Math. 29 (1979), 295–303.Google Scholar
12. Marshall, M., Spaces of orderings IV, Can. J. Math. 32 (1980), 603–627.Google Scholar
13. Marshall, M., Abstract Witt rings, Queen's papers in pure and applied Math. 57, Queen's Univ. (1980).Google Scholar
14. Marshall, M. and Yucas, J., Linked quaternionic mappings and their associated Witt rings, Pac. J. Math. 95 (1981).Google Scholar
15. Szczepanik, L., Fields and quadratic form schemes with the index of the radical not exceeding 16, preprint.Google Scholar
16. Szczepanik, L., Quadratic form schemes with non trivial radical, preprint.Google Scholar
17. Witt, E., Théorie der Quadratischen Formen in beliebigen Kôrpern, J. reine angew. Math. 176 (1937), 31–44.Google Scholar
18. Yucas, J., A note on quadratic forms, rigid elements, and orderings of fields, preprint.Google Scholar