Re: Galois subextensions question

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

Bill,

I might not have stated the problem correctly. Say I have a degree n 
polynomial and want to compute the fixed field by the A_n subgroup? Or an 
S_m subgroup (m<n), etc. (you can give the subgroup in any way you want).

I didn't mean bnfinit, but you need to run first galoisinit(), which needs 
nfinit... I do not want to compute the Galois closure group, but a fixed 
subextension.

Ariel

On Tue, 1 Oct 2013, Bill Allombert wrote:

> On Mon, Sep 30, 2013 at 08:07:37PM -0500, Ariel Pacetti wrote:
> > 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).
>
> It all depends on how your subgroup G is given.
>
> > 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.
>
> Which bnfinit ? I have a script that lets you compute the Galois closure
> of a field assuming the closure is of degree less than 10000, says.
>
> Cheers,
> Bill.