# Field of Iterated Laurent Series and its Brauer Group

## TL;DR

This paper investigates the symbol length in the Brauer group of iterated Laurent series fields over algebraically closed fields, revealing differences based on characteristic and providing new insights into quadratic forms.

## Contribution

It establishes the symbol length as n-1 in characteristic p, contrasting with known results in characteristic not p, and explores properties of Pfister forms over these fields.

## Key findings

- Symbol length in characteristic p is n-1.
- Pairs of Pfister forms may lack an (n-1)-fold factor.
- Differences in symbol length depend on the field's characteristic.

## Abstract

The symbol length of ${_pBr}(k(\!(\alpha_1)\!)\dots(\!(\alpha_n)\!))$ for an algebraically closed field $k$ of $\operatorname{char}(k) \neq p$ is known to be $\lfloor \frac{n}{2} \rfloor$. We prove that the symbol length for the case of $\operatorname{char}(k) = p$ is rather $n-1$. We also show that pairs of anisotropic quadratic or bilinear $n$-fold Pfister forms over this field need not share an $(n-1)$-fold factor.