Symbol lengths in Milnor $K$-theory

Let $F$ be a field and $p$ a prime number. The $p$-symbol length of $F$, denoted by $\lambda_p(F)$, is the least integer $l$ such that every element of the group $K_2 F/p K_2F$ can be written as a sum of $\leq l$ symbols (with the convention that $\lambda_p(F)=\infty$ if no such integer exists). In this article, we obtain an upper bound for $\lambda_p(F)$ in the case where the group $F^\times/{F^\times}^p$ is finite of order $p^m$. This bound is $\lambda_p(F)\leq \frac{m}{2}$, except for the case where $p=2$ and $F$ is real, when the bound is $\lambda_2(F)\leq \frac{m+1}{2}$. We further give examples showing that these bounds are sharp.

Keywords

Milnor K-groups, symbols, symbol length, power norm residue algebra, symbol algebra, quadratic forms, level, Brauer group, Merkurjev-Suslin theorem

2010 Mathematics Subject Classification

Primary 19D45. Secondary 11E04, 11E81, 16K50.

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Published 1 January 2004