The $u^n$-invariant and the Symbol Length of $H_2^n(F)$

Abstract

Given a field $F$ of $\mathrm{char}(F)=2$, 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 $s{l}_2^n(F)$. We show for any $n\ge 2$ that if $2^n\le u^n(F)\le u^2(F)<\infty$ then $s{l}_2^n(F)\le {\prod }_{i=2}^n(\frac{u^i(F)}{2}+1-2^{i-1})$. As a result, if $u(F)$ is finite then $s{l}_2^n(F)$ is finite for any $n$, a fact which was previously proven when $\mathrm{char}(F)\ne 2$ by Saltman and Krashen. We also show that if $s{l}_2^n(F)=1$ then $u^n(F)$ is either $2^n$ or $2^{n+1}$.