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# Homology, Homotopy and Applications

## Volume 6 (2004)

### Number 1

### Symbol lengths in Milnor $K$-theory

Pages: 17 – 31

DOI: http://dx.doi.org/10.4310/HHA.2004.v6.n1.a3

#### Authors

#### Abstract

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.