This is a follow-up to the following answer:


[Solvable class field theory](https://mathoverflow.net/questions/4379/solvable-class-field-theory/4386#4386)


in which it is stated as a "folklore" conjecture that the maximal solvable extension of Q is pseudo algebraically closed (this means, in particular, any geometrically connected variety over Q has a point over a solvable extension).


I am curious what evidence there is to support such a conjecture.


In addition, what can be said for the analogous statement for global function fields?



Evidence for $Q^{\operatorname{solv}}$ being pseudo-algebraically-closed
So far as I know, there is no compelling evidence to support this conjecture. (And some leading arithmetic geometers think it is false.) Rather, there are some very interesting consequences of this conjecture, e.g. a solution of the Inverse Galois Problem over Q^{solv}: in other words, for any finite group G, there exists a tower of radical extensions 


K\_0 = Q, K\_1 = K\_0(a\_0^{1/n\_0}) < K\_2 = K\_1(a\_1^{1/n\_1}) < ... <= K\_n


and a Galois extension L/K\_n with Galois group isomorphic to G. It also shows that geometrically irreducible algebraic varieties "acquire rational points" in a very different way from irreducible zero-dimensional varieties (which have a unique minimal 
splitting field which need not be solvable). 


Of course, interesting things which follow from a conjecture are, if anything, evidence *against* the truth of the conjecture, although they support the claim that the *question* is interesting. 


There is one impressive result towards this conjecture, namely the Ciperiani-Wiles theorem: let C\_{/Q} be a genus one curve with points everywhere locally and semistable Jacobian elliptic curve E. Then C(Q^{solv}) is nonempty.


On the negative side, there is a paper of Ambrus Pal which constructs, for each sufficiently large integer g, a curve C of genus g over a field K which does not admit any points over the maximal solvable extension of K. (Here K is not a number field.)


On the other hand, as far as I know, it is still open to find an absolutely irreducible variety V/Q which fails to have rational points over the maximal *metabelian extension* of Q, i.e., over (Q^{ab})^{ab}. For some thoughts about this, see


<http://alpha.math.uga.edu/~pete/abeliantalk.pdf>


ADDENDUM:


I forgot to address the last part of the question: what about global function fields?


As I alluded to above, there are counterexamples over the function field of a sufficiently complicated ground field, like Q. Of course you mean a finite extension of F\_q(T), in which case I think absolutely nothing is known. In particular, I believe the analogue of Ciperiani-Wiles is open here, and might not be a straightforward adaptation, since C-W uses results on the modularity of elliptic curves. This could make a nice thesis problem...but I would talk to Mirela Ciperiani before doing any serious work on it.


