Let $A$ be a principal homogeneous space for a commutative algebraic group defined over a field $k$ that contains all roots of unity. Is it true that $A$ has a $K$-point for an extension $K \supset k$ such that $\mathrm{Gal}(K/k)$ is Abelian?

Çiperiani and Wiles had to work hard to prove that torsors for elliptic curves over the rationals had points in solvable extensions. That makes me think that the answer to your question is no, otherwise their result would easily follow.
– Felipe VolochMay 28 '13 at 15:48

Pete Clark has a paper on abelian points of varieties where he gives examples to show that they don't always exist, although I don't know for sure if he gives any examples with fields like your $k$.
– Keenan KidwellMay 28 '13 at 15:50

Over a $C_1$ field of characteristic zero, Springer showed that every phs for a connected algebraic group has a point. It has been conjectured by Artin that algebraic extensions of $Q$ containing all roots of 1 are $C_1$ (see Serre's Corps Locaux X, Section 7), but this (I think) is not known.
– anonMay 28 '13 at 17:18

@anon, the C_1 situation is somewhat degenerate. I am interested in the situation when there is no necessarily a rational point from the field of definition, but there is a point from an Abelian extension of it.
– Dima SustretovMay 29 '13 at 7:40

Dear Felipe, this makes a convincing point indeed. Do you know of any specific counterexample to illustrate the failure? I had a (brief) look at Ciperiani and Wiles article but couldn't extract one. Can one hope for the statement to hold (non-trivially) for some class of fields, maybe for function fields?
– Dima SustretovMay 29 '13 at 7:52