Sums of two symbols in $K_2(F)/2K_2(F)$ in characteristic two

Abstract

In this paper, study sums $A=\{a,b{\}}_2+\{c,d{\}}_2$ of two symbols in $K_2(F)/2K_2(F)$ when $\mathrm{char}(F)=2$. We first prove a chain lemma that connects $A$ to $B=\{\hat{I}\pm ,{\hat{I}}^2{\}}_2+\{{\hat{I}}^3,\hat{I}´{\}}_2$ by a finite sequence of small steps when $A\equiv B$. We use this lemma to prove that $\{a,b,c,d{\}}_2\in K_4(F)/2K_4(F)$ is a well-defined invariant of $A$, and that this invariant is trivial if and only if $A$ is congruent to a single symbol in $K_2(F)/4K_2(F)$. We also bound the symbol length of $C$ in $K_2(F)/2^m{K}_2(F)$ from above when $C$ is the sum of up to four symbols in $K_2(F)/2^{m+1}{K}_2(F)$.