# Linkage of Pfister forms over $\mathbb{C}(x_1,\ldots,x_n)$

## TL;DR

This paper constructs a specific set of Pfister forms over the complex rational function field that do not share a common factor, answering a previously open question negatively.

## Contribution

It demonstrates the existence of $n$-fold Pfister forms over $ ext{C}(x_1, ext{...},x_n)$ without a common $(n-1)$-fold factor, using valuation theory and recent bilinear form results.

## Key findings

- Existence of such Pfister forms over complex rational function fields.
- Negative answer to Becher's question about common factors.
- Application of dyadic valuation and symmetric bilinear form theory.

## Abstract

In this note, we prove the existence of a set of $n$-fold Pfister forms of cardinality $2^n$ over $\mathbb{C}(x_1,\dots,x_n)$ which do not share a common $(n-1)$-fold factor. This gives a negative answer to a question raised by Becher. The main tools are the existence of the dyadic valuation on the complex numbers and recent results on symmetric bilinear over fields of characteristic 2.