Fields of u-invariant 9

• 成果类型: Article
• 署名作者: Izhboldin, OT
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/3062141
• 发表日期: 2001
• 页码: 529-587
• 关键词: quadratic-forms; cohomological invariants; unramified cohomology; splitting patterns; K-THEORY; dimensions; ALGEBRAS; quadrics

摘要

Let F be a field of characteristic not equal 2. The u-invariant of the field F is defined as the maximum dimension of anisotropic quadratic forms over F. It is well-known that the u-invariant cannot be equal to 3, 5, or 7. We construct a field F with u-invariant 9. It is a first example of a field with odd u-invariant > 1. The proof uses the computation of the third Chow group of projective quadrics X-phi corresponding to quadratic forms phi. We compute CH3(X-phi) for all phi except for the case dim phi = 8. In our computation we use results of B. Kahn, M. Rost, and R. Sujatha on unramified cohomology and the third Chow group of quadrics ([23]). We compute unramified cohomology H-nr(4)(F(phi)/F) for all forms phi of dimension greater than or equal to 9. We apply our results to prove several conjectures. In particular, we prove a conjecture of Bruno Kahn on the classification of forms of height 2 and degree 3 for all fields of characteristic zero.