Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-20T07:29:32.226Z Has data issue: false hasContentIssue false
  • English
  • Français

Abstract Witt Rings When Certain Binary Forms Represent Exactly Four Elements

Published online by Cambridge University Press:  20 November 2018

Craig M. Cordes
Craig M. Cordes*
Affiliation:
Department of Mathematics Louisiana State University Baton Rouge, Louisiana 70803 U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

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.

An abstract Witt ring (R, G) of positive characteristic is known to be a group ring S[Δ] with ﹛1﹜ ≠ Δ ⊆ G if and only if it contains a form〈1,x〉, x ≠1, which represents only the two elements 1 and x. Carson and Marshall have characterized all Witt rings of characteristic 2 which contain binary forms representing exactly four elements. Such results which show R is isomorphic to a product of smaller rings are helpful in settling the conjecture that every finitely generated Witt ring is of elementary type. Here, some special situations are considered. In particular if char(R) = 8, |D〈l, 1〉| = 4, and R contains no rigid elements, then R is isomorphic to the Witt ring of the 2-adic numbers. If char(R) = 4, |D〈l,a〉| = 4 where a ∈ D〈1, 1〉, and R contains no rigid elements, then R is either a ring of order 8 or is the specified product of two Witt rings at least one of which is a group ring. In several cases R is realized by a field.

Keywords

11E81
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 6 , 01 December 1993 , pp. 1184 - 1199
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Berman, L., Cordes, C. and Ware, R., Quadratic forms, rigid elements, and formal power series fields, J. Algebra 66(1980), 123–133.Google Scholar
2. Bos, R., Quadratic forms, orderings, and abstract Witt rings, Dissertation, Utrecht (1984).Google Scholar
3. Carson, A. and Marshall, M., Decomposition of Witt rings, Can. J. Math. 34(1982), 1276.1302.Google Scholar
4. Cordes, C., The Witt group and the equivalence of fields with respect to quadratic forms, J. Algebra 26(1973), 400–421.Google Scholar
5. Cordes, C., Kaplansky's radical and quadratic forms over non-real fields, Acta Arith. 28(1975), 253–261.Google Scholar
6. Kula, M., Fields with prescribed quadratic form schemes, Math. Zeit. 167(1979), 202–212.Google Scholar
7. Kula, M., Szczepanik, L. and Szymiczek, K., Quadratic form schemes and quaternionic schemes, Fund. Math. 130(1988), 181–190.Google Scholar
8. Marshall, M., Abstract Witt rings, Queen's Papers in pure and applied Math. 57, Queen's Univ. (1980).Google Scholar
9. Marshall, M., Decomposing Witt rings of characteristic two, Rocky Mountain J. Math. 19(1989), 793–806.Google Scholar
10. Szczepanik, L., Quadratic form schemes with non-trivial radical, Colloquium Math. 49(1985), 143–160.Google Scholar