On the nonexcellence of field extensions $F(\pi)/F$

Abstract

For any $n\ge3$, we construct a field $F$ and an $n$-fold Pfister form $\varphi$ such that the field extension $F(\varphi)/F$ is not excellent. We prove that $F(\varphi)/F$ is universally excellent if and only if $\varphi$ is a Pfister neighbor of dimension $\le4$.