#### V. Chernousov, Ph. Gille, Z. Reichstein, *
Resolving G-torsors by abelian base extensions,
* Journal of Algebra, 296 (2006), 561--581.

**Abstract:**
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 H^1(K, S) -> H^1(K, G)
is surjective for every field extension K/k.
We give two applications of this result
in the case where k an algebraically closed
field of characteristic zero and K/k is
finitely generated. First we show
that for every z in H^1(K, G)
there exists an abelian field extension L/K
such that z_L \in H^1(L, G) is represented
by a G-torsor over a projective variety.
In particular, we prove that z_L has trivial
point obstruction.
As a second application of our main theorem,
we show that a (strong) variant
of the algebraic form of Hilbert's 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 Koenigsmann, answer
a question of Tits and establish
an important case of Serre's
Conjecture II for the group E_8.

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