Resolving G-torsors by abelian base extensions

Let G be a linear algebraic group defined over a field k. We prove that, under mild assumptions on k and G, there exists a finite k-subgroup S of G such that the natural map H1(K,S)→H1(K,G) is surjective for every field extension K/k. We give several applications of this result in the case where k an algebraically closed field of characteristic zero and K/k is finitely generated. In particular, we prove that for every α H1(K,G) there exists an abelian field extension L/K such that αL H1(L,G) is represented by a G-torsor over a projective variety. From this we deduce that αL has trivial fixed point obstruction. We also show that a (strong) variant of the algebraic form of Hilbertʹs 13th problem implies that the maximal abelian extension of K has cohomological dimension 1. The last assertion, if true, would prove conjectures of Bogomolov and Königsmann, answer a question of Tits and establish an important case of Serreʹs Conjecture II for the group E8.