Conjecture I for unirational algebraic groups over imperfect fields

Abstract

Serre's Conjecture I states that the first Galois cohomology set of any smooth connected linear algebraic group is trivial over a perfect field of cohomological dimension at most 1. We prove that this result remains valid for any unirational algebraic group, dropping the perfection assumption on the field. To do so, we rely on the theory of pseudo-reductive groups, combined with the structure of unirational wound unipotent groups and the recent theory of permawound unipotent groups. Finally, we extend several related results on Galois cohomology associated with the Conjecture.