Quadratic Forms and u-invariant

Please enable javascript in your browser.

Khajehvand, Bahador | 2016

772 Viewed

Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 48606 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Gholamzadeh Mahmoudi, Mohammad
Abstract:
For a field of characteristic not two, the classical u-invariant is defined as the maximal dimension of anisotropic quadratic forms over F. Initially Kaplansky conjectured that u(F), when finite, is always a 2-power. Later Merkurjev constructed a field F such that u(F) = 6. This dissertation examines in detail the article: R. Elman, T. Y. Lam, Quadratic forms and the u-invariant. I. Math. Z. 131, 283-304 (1973). in which the notion of ”generalized u-invariant” (motivated by Pfister’s Local-Global Principle) was defined as the maximal dimension of anisotropic torsion quadratic forms over F. This is indeed a right generalization of the definition of the classical u-invariant since it not only coincides with the old one when F is nonreal but also contains additional information when F is formally real. One of the well-known inequalities about the u-invariant when the field F is nonreal is the inequality u q, where q denotes the number of square classes of F. This result is due to M. Kneser. In the work, various generalizations of this inequality are provided in the context of general u-invariant
Keywords:
Quadratic Forms ; Witt Ring ; Pfister Forms ; Formally Real Field ; Anisotropic Form ; Binary Form