Types of Linkage of Quadratic Pfister Forms

TL;DR

This paper investigates the linkage properties of quadratic Pfister forms over fields of positive characteristic, establishing conditions under which certain symbols are trivial and constructing examples of linked Pfister forms.

Contribution

It proves a new triviality result for symbols sharing common factors and describes methods to construct linked Pfister forms with shared factors.

Findings

Shared factors imply triviality of certain symbols in cohomology.

All totally separably linked 2-fold quadratic Pfister forms are inseparably linked.

Methods to construct non-isomorphic Pfister forms sharing common factors.

Abstract

Given a field $F$ of positive characteristic $p$, $\theta \in H_p^{n-1}(F)$ and $\beta,\gamma \in F^\times$, we prove that if the symbols $\theta \wedge \frac{d \beta}{\beta}$ and $\theta \wedge \frac{d \gamma}{\gamma}$ in $H_p^n(F)$ share the same factors in $H_p^1(F)$ then the symbol $\theta \wedge \frac{d \beta}{\beta} \wedge \frac{d \gamma}{\gamma}$ in $H_p^{n+1}(F)$ is trivial. We conclude that when $p=2$, every two totally separably $(n-1)$-linked $n$-fold quadratic Pfister forms are inseparably $(n-1)$-linked. We also describe how to construct non-isomorphic $n$-fold Pfister forms which are totally separably (or inseparably) $(n-1)$-linked, i.e. share all common $(n-1)$-fold quadratic (or bilinear) Pfister factors.