Types of Linkage of Quadratic Pfister Forms

Abstract

Given a field $F$ of positive characteristic $p$, $\hat{I}¸\in H_p^{n-1}(F)$ and ${\hat{I}}^2,{\hat{I}}^3\in F^{\times }$, we prove that if the symbols $\hat{I}¸\wedge {\frac{d{\hat{I}}^2}{\hat{I}}}^2$ and $\hat{I}¸\wedge {\frac{d{\hat{I}}^3}{\hat{I}}}^3$ in $H_p^n(F)$ share the same factors in $H_p^1(F)$ then the symbol $\hat{I}¸\wedge {\frac{d{\hat{I}}^2}{\hat{I}}}^2\wedge {\frac{d{\hat{I}}^3}{\hat{I}}}^3$ 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.