Differential Forms, Linked Fields and the $u$-Invariant

Abstract

We associate an Albert form to any pair of cyclic algebras of prime degree $p$ over a field $F$ with $\mathrm{char}(F)=p$ which coincides with the classical Albert form when $p=2$. We prove that if every Albert form is isotropic then $H^4(F)=0$. As a result, we obtain that if $F$ is a linked field with $\mathrm{char}(F)=2$ then its $u$-invariant is either $0,2,4$ or $8$.