论文标题
Milnor $ k $ -groups中的符号长度
Symbol Length of Classes in Milnor $K$-groups
论文作者
论文摘要
给定一个字段$ f $，一个积极的整数$ m $和一个整数$ n \ geq 2 $，我们证明了米尔诺的$ k $ -groups $ k $ -groups $ k_n f/2^m k_n f $等效于单个符号的单个符号的符号长度，该符号与$ k_n f/k_n f/2^{m+1} $ 2 $ 2 $} $ f \supseteqμ_{2^{m+1}} $。由于$ n = 2 $，$ k_2 f/2^m k_2 f \ cong {_ {2^m} br（f）} $，这与$ 2 $的上限同时发生$ 2 $，符号的符号长度为$ 2^m $的中央代数$ 2^m $，这是brauer的单个符号algebra of Single Algebra of级$ 2.嵌入到$ k_n f/2^{m+1} k_n f $的情况下是符号长度2、3和4（后者时为$ n = 2 $）。我们结束了研究$ k_3/3^m k_3 f $的类符号长度的符号长度，其嵌入$ k_3 f/3^{m+1} k_3 f $是$ f \ f \supSeteeqμ__{3^^{m+1}} $时的一个符号。
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 $F \supseteq μ_{2^{m+1}}$. 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 $F \supseteq μ_{3^{m+1}}$.
