## Homology, Homotopy and Applications

- Homology Homotopy Appl.
- Volume 6, Number 1 (2004), 17-31.

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

Karim Johannes Becher and Detlev W. Hoffmann

#### 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.

#### Article information

**Source**

Homology Homotopy Appl., Volume 6, Number 1 (2004), 17-31.

**Dates**

First available in Project Euclid: 13 February 2006

**Permanent link to this document**

https://projecteuclid.org/euclid.hha/1139839542

**Mathematical Reviews number (MathSciNet)**

MR2061565

**Zentralblatt MATH identifier**

1069.19004

**Subjects**

Primary: 19D45: Higher symbols, Milnor $K$-theory

Secondary: 11E04: Quadratic forms over general fields 11E81: Algebraic theory of quadratic forms; Witt groups and rings [See also 19G12, 19G24]

#### Citation

Becher, Karim Johannes; Hoffmann, Detlev W. Symbol lengths in Milnor $K$-theory. Homology Homotopy Appl. 6 (2004), no. 1, 17--31. https://projecteuclid.org/euclid.hha/1139839542