Symbol Length of Classes in Milnor $K$-groups

Abstract

Given a field $F$, a positive integer $m$ and an integer $n\geq 2$, we prove that the symbol length of classes in Milnor's $K$-groups $K_n F/2^m K_n F$ that are equivalent to single symbols under the embedding into $K_n F/2^{m+1} K_n F$ is at most $2^{n-1}$ under the assumption that $F \supseteq \mu_{2^{m+1}}$. Since for $n=2$, $K_2 F/2^m K_2 F \cong {_{2^m}Br(F)}$, this coincides with the upper bound of $2$ for the symbol length of central simple algebras of exponent $2^m$ that are Brauer equivalent to a single symbol algebra of degree $2^{m+1}$ proved by Tignol in 1983. We also consider the cases where the embedding into $K_n F/2^{m+1} K_n F$ is of symbol length 2, 3 and 4 (the latter when $n=2$). We finish with studying the symbol length of classes in $K_3/3^m K_3 F$ whose embedding into $K_3 F/3^{m+1} K_3 F$ is one symbol when $F \supseteq \mu_{3^{m+1}}$.