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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
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Abstract

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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

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