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

TL;DR

This paper investigates sums of two symbols in the second K-group modulo 2 in characteristic two, establishing a chain lemma, defining a new invariant, and bounding symbol length in higher powers.

Contribution

Introduces a chain lemma linking sums of symbols, defines a new invariant in K-theory, and provides bounds on symbol length in characteristic two fields.

Findings

Proves a chain lemma connecting sums of symbols via small steps.

Shows the invariant is trivial iff the sum is congruent to a single symbol.

Provides bounds on symbol length in higher K-groups.

Abstract

In this paper, study sums $A=\{a,b\}_2+\{c,d\}_2$ of two symbols in $K_2(F)/2K_2(F)$ when $\operatorname{char}(F)=2$. We first prove a chain lemma that connects $A$ to $B=\{\alpha,\beta\}_2+\{\gamma,\delta\}_2$ 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 $K_2(F)/2^{m+1}K_2(F)$.