Fields of $u$-invariant $9$


Let $F$ be a field of characteristic $\ne 2$. 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 $\mathrm{CH}^3(X_\phi)$ 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 $\ge 9$. 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.



Oleg T. Izhboldin