Ariel Pacetti on Tue, 01 Oct 2013 15:25:16 +0200

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Re: Galois subextensions question


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.


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.