Oleg T. Izhboldin: Fields of u-invariant 9

oleg@mathematik.uni-bielefeld.de

Submission: 2000, Feb. 25

Let $F$ be a field of characteristic $\ne2$. The $u$-invariant of the field $F$ is defined as the maximal 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 the 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 $\CH^3(X_\phi)$ completely except for the case $\dim\phi=8$. In our computation we use the results of B.~Kahn, M.~Rost, and R.~Sujatha on the unramified cohomology and the third Chow group of quadrics (\cite{Unram I}). We compute the unramified cohomology $H^4_{nr}(F(\phi)/F)$ for all forms of dimension $\ge9$. 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.

1991 Mathematics Subject Classification:

Keywords and Phrases: Quadratic form, Chow groups, unramified cohomology

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