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Submission: 2004, Mar 29
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 for any field extension K/k, the G-torsors over K come from S-torsors over 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. The first one is about making a G-torsor strongly unramified after an abelian base extension; the second one deals with the maximal abelian extension of K in view of a strong variant of Hilbert 13-th problem.
2000 Mathematics Subject Classification: 11E72, 14L30, 14E20
Keywords and Phrases: Linear algebraic group, torsor, unramified torsor, non-abelian cohomology, G-cover, cohomological dimension, Brauer group, group action, fixed point obstruction
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