Symbol length of classes in Milnor 𝐾-groups

Abstract

Given a field F F , a positive integer m m and an integer n ≥ 2 n\geq 2 , it is proved that the symbol length of classes in Milnor’s K K -groups K n F / 2 m K n F 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 K_n F/2^{m+1} K_n F is at most 2 n − 1 2^{n-1} under the assumption that F ⊇ μ 2 m + 1 F \supseteq \mu _{2^{m+1}} . Since K 2 F / 2 m K 2 F ≅ 2 m B r ( F ) K_2 F/2^m K_2 F \cong {_{2^m}Br(F)} for n = 2 n=2 , this coincides with the upper bound of 2 2 (proved by Tignol in 1983) for the symbol length of central simple algebras of exponent 2 m 2^m that are Brauer equivalent to a single symbol algebra of degree 2 m + 1 2^{m+1} . The cases where the embedding into K n F / 2 m + 1 K n F K_n F/2^{m+1} K_n F is of symbol length 2 2 , 3 3 , and 4 4 (the last when n = 2 n=2 ) are also considered. The paper finishes with the study of the symbol length for classes in K 3 / 3 m K 3 F K_3/3^m K_3 F whose embedding into K 3 F / 3 m + 1 K 3 F K_3 F/3^{m+1} K_3 F is one symbol when F ⊇ μ 3 m + 1 F \supseteq \mu _{3^{m+1}} .