Galois subextensions question

[Ariel Pacetti <apacetti@math.utexas.edu>]

Dear Pari users,

suppose I start with a degree n (say n small) polynomial for which I know 
the Galois closure is the whole S_n (this is not needed, but to easy the 
question). Then if I take any subgroup of S_n, there is an extension of Q 
fixed by this subgroup. Is there a way to compute such extension in GP 
without computing the whole Galois closure? (this is of huge dimension!). 
I couldn't find anything in this direction (since the original extension 
is not Galois).

Here is a "heuristic" and probably not efficient way to do it, take a 
formal basis of the extension in terms of succesive roots of the 
polynomial, and then the fixed field is just the vector space of solutions 
of a linear system. A random solution will generate the extension (over 
Q). Then one can compute its minimal polinomial (formally or using 
numerical approximation as a complex number + algdep). I am thinking of a 
degree 5 or 6 polynomial, where the space is 120 or 720 dimentional, so 
the linear algebra should work, but the bnfinit won't.

If anyone knows something in this direction will be of great help (before 
writing some code to compute the linear system).

Ariel