Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-27T09:14:53.966Z Has data issue: false hasContentIssue false
  • English
  • Français

Indecomposable Positive Quadratic Forms

Published online by Cambridge University Press:  20 November 2018

Martin Krüskemper
Martin Krüskemper*
Affiliation:
Mathematisches Institut der Universität Einsteinstraβe 62 D-4400 Münster
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.

Let F be a formally real field. A quadratic form q is called positive if sgnp ≧ 0 for all orderings P of F. A positive q is called decomposable if there exist positive forms q1, q2 such that q = q1⊥q2. Otherwise it is called indecomposable. In a first part we ask for which F there exist indecomposable three dimensional forms over F. We show that such forms exist iff F does not satisfy the property (A) defined in (J. K. Arason, A. Pfister: Zur Théorie der quadratischen Formen über formal reellen Körpern, Math Z. 153, 289-296 (1977)). We use an indecomposable three dimensional form defined by Arason and Pfister to construct indecomposable forms of arbitrary dimension. Then we examine the question for which fields F every positive form over F represents a nonzero sum of squares.

Keywords

11E04
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 33 , Issue 3 , 01 September 1990 , pp. 305 - 310
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Arason, J. K., A. Pfister, Zur Théorie der quadratischen Formen über formal reellen Körpern, Math, Z. 153 (1977), 289–296.Google Scholar
2. Bröcker, L., Characterization of fans and hereditarily pythagorean fields, Math. Z. 151 (1976), 149— 163.Google Scholar
3. Carson, A. B., Marshall, M., Decomposition of Witt rings, Can. J. Math. 34 (1986), 1276–1302.Google Scholar
4. Cordes, C., Quadratic forms over nonformally real fields with a finite number of quaternion algebras, Pac. J. Math. 63 (1976), 357–365.Google Scholar
5. Krüskemper, M., Scharlau, W., On positive quadratic forms, Bull. Soc. Math. Belg. 40 ser. A, (1988), 255–280.Google Scholar
6. Kula, M., Fields and quadratic form schemes, Pr. Nauk Uniw. Slask. Katowicach 693, Ann. Math. Sil. 1 (1985), 7–22.Google Scholar
7. Lam, T. Y., Orderings, valuations and quadratic forms, Am. Math. Soc. Regional Conf. Series Math. No 52. 1983.Google Scholar
8. Marshall, M., Classification of finite spaces of orderings, Canad. J. Math. 31 (1979), 320–330.Google Scholar
9. Marshall, M., Spaces of orderings IV, Canad. J. Math. 32 (1980), 603–627.Google Scholar
10. Prestel, A., Remarks on the Pythogoras and Hasse number of real fields, J. Reine Angew. Math. 303/304 (1978), 284–294.Google Scholar
11. Prestel, A., R. Ware, Almost isotropic quadratic forms, J. London Math. Soc. 19 (1979), 241–244.Google Scholar
12. Scharlau, W., Quadratic and Hermitian forms, Berlin, Heidelberg, New York, Springer 1985.Google Scholar