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

TL;DR

This paper introduces a new association between Albert forms and pairs of cyclic algebras over fields of prime characteristic, linking isotropy of Albert forms to the vanishing of a cohomology group and deriving bounds on the u-invariant for linked fields.

Contribution

It defines a generalized Albert form for prime characteristic fields and establishes a connection between isotropy of these forms and the vanishing of the fourth Galois cohomology group.

Findings

If all Albert forms are isotropic, then H^4(F)=0.

For linked fields of characteristic 2, the u-invariant is limited to 0, 2, 4, or 8.

Abstract

We associate an Albert form to any pair of cyclic algebras of prime degree $p$ over a field $F$ with $\operatorname{char}(F)=p$ 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 $\operatorname{char}(F)=2$ then its $u$-invariant is either $0,2,4$ or $8$.