Bill Allombert on Tue, 01 Oct 2013 14:03:10 +0200

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Re: Galois subextensions question

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.