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

TL;DR

This paper investigates the relationship between the $u^n$-invariant and the symbol length of $H_2^n(F)$ over fields of characteristic 2, establishing bounds and finiteness conditions that extend previous results to characteristic 2.

Contribution

It introduces new bounds linking the $u^n$-invariant and symbol length of $H_2^n(F)$ in characteristic 2 fields, extending known results from characteristic not 2.

Findings

If $2^n \\leq u^n(F) \\leq u^2(F) < \\infty$, then $sl_2^n(F)$ is bounded by a product involving $u^i(F)$.

Finiteness of $u(F)$ implies finiteness of $sl_2^n(F)$ for all $n$ in characteristic 2.

When $sl_2^n(F)=1$, then $u^n(F)$ is either $2^n$ or $2^{n+1}$.

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 $n \geq 2$ that if $2^n \leq u^n(F) \leq u^2(F) < \infty$ then $sl_2^n(F) \leq \prod_{i=2}^n (\frac{u^i(F)}{2}+1-2^{i-1})$. 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 $2^{n+1}$.