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arXiv · 1701.01367

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

Adam Chapman·Andrew Dolphin
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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 $8$.

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Adam Chapman, Andrew Dolphin. 2017-01-05. Differential Forms, Linked Fields and the $u$-Invariant. https://arxiv.org/abs/1701.01367

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