Types of Linkage of Quadratic Pfister Forms

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$H_p^n(F)$sharethesamefactorsin$H_p^1(F)$thenthesymbol$\theta \wedge \frac{d \beta}{\beta} \wedge \frac{d \gamma}{\gamma}$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.