|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. Cheers, Bill.