On a property of special groups

Zinovy Reichstein and Boris Youssin

Let G be an algebraic group defined over an algebraically closed field k of characteristic zero. We give a simple proof of the following result: if H¹(K₀, G) = {1} for some finitely generated field extension K₀/k of transcendence degree \geq 3 then H¹(K, G) = {1} for every field extension K/k.