Abstract: Let G be an algebraic group defined over an algebraically closed field k of characteristic zero. We show that if H^1(K, G) = { 1 } for some finitely generated field extension K_0/k of transcendence degree > 2 then H^1(L, G) = { 1 } for every field extension L/k. This shows that Serre's conjectures I and II have no analogues for fields of cohomological dimentsion > 2.