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

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符号长度类在米尔诺𝐾-groups

给定一个域F、一个正整数m和一个整数n≥2n\geq2，证明了Milnor的K-群KnF/2mKnFK_nF/2^mK_nF中等价于嵌入到KnF/2中的单个符号的类的符号长度m+1 K n F K_ n F/2^｛m+1｝K_ n F在假定Fμ2 m+1 F\supseteqμ_｛2^｛m+1｝｝。由于当n=2n=2时，K2 F/2 m K2 FŞ2 m Br（F）K_，这与指数为2m2^m的中心单代数的符号长度的2 2的上界（由Tignol在1983年证明）一致，该中心单代数是Brauer等价于2 m+12^{m+1}度的单符号代数。还考虑了嵌入到K n F/2 m+1 K n F K_n F/2^｛m+1｝K_n F中的符号长度为2 2、3 3和4 4（最后当n=2 n=2时）的情况。本文最后研究了

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