The $u^n$-invariant and the symbol length of $H_2^n(F)$

Abstract:

Given a field $F$ of $\operatorname {char}(F)=2$, we define $u^n(F)$ to be the maximal dimension of an anisotropic form in $I_q^n F$. For $n=1$ it recaptures the definition of $u(F)$. We study the relations between this value and the symbol length of $H_2^n(F)$, denoted by $sl_2^n(F)$. We show for any $n \geqslant 2$ that if $2^n \leqslant u^n(F) \leqslant u^2(F) < \infty$, then $sl_2^n(F) \leqslant \prod _{i=2}^n (\frac {u^i(F)}{2}+1-2^{i-1})$. As a result, if $u(F)$ is finite, then $sl_2^n(F)$ is finite for any $n$, a fact which was previously proven when $\operatorname {char}(F) \neq 2$ by Saltman and Krashen. We also show that if $sl_2^n(F)=1$, then $u^n(F)$ is either $2^n$ or $2^{n+1}$.