Common Slots of Bilinear and Quadratic Pfister Forms

TL;DR

The paper demonstrates that over fields of characteristic 2 with 2-rank n, there exist 2^n bilinear and quadratic Pfister forms with no common slots, answering a previously open question negatively.

Contribution

It establishes the existence of multiple Pfister forms with no common slots over certain fields, extending understanding of their structure and relationships.

Findings

Existence of 2^n bilinear Pfister forms with no common slot

Analogous result for quadratic Pfister forms

Answers Becher's question negatively

Abstract

We show that over any field $F$ of $\operatorname{char}(F)=2$ and 2-rank $n$, there exist $2^n$ bilinear $n$-fold Pfister forms that have no slot in common. This answers a question of Becher's in the negative. We provide an analogous result also for quadratic Pfister forms.