"Jordi Guti?rrez Hermoso" <jordigh at gmail.com> writes:
> Let's move the discussion back to the list...
>
> On 24/01/2008, Hugo Coolens <hugocoolens at skynet.be> wrote:
>> The thing is, I have a whole family of polynomials related to an
>> electrical network and I'd love to see a general method to solve
>> them (factorize or determine the roots) starting from the coefficients,
>> I don't know much about Galois theory but it would be great just to see
>> it work on an example like I sent you, maybe then I'll get the necessary
>> insight
>
>
> This doesn't seem like an easy question in general. Of course if you
> can find enough roots by radicals until your polynomial is of degree 4
> or less, then you have found them all. In general, I think we would
> first need routines for computing Galois groups of polynomials of
> small degrees. Last time I checked, Maple had routines for computing
> Galois groups of polynomials of degree up to 9, but I don't know if
> Maxima has any such thing.
>
> Using tricks such as computing discriminants and other assorted
> methods, it's possible to automate the computation of the Galois group
> for polynomials. A decomposition series for the Galois group, if it
> exists (i.e. if it's soluble), can then give a clue on what radicals
> the solution will be, although I don't have details for this.
>
> I would have to grep the literature to see if I can find anything
> about computer algorithms for solutions by radicals or determine that
> they're impossible. It's an interesting problem, and I'm sure someone
> else has already solved it or worked on it extensively.
>
> Anyone else, please feel free to contribute further thoughts.
>
GAP has functions for computing the Galois group of a polynomial.
For example
gap> x := Indeterminate(Rationals);;
gap> GaloisType(x^10-5*x^4-4);
37
gap> g := TransitiveGroup(10, last);
[2^4]S(5)
gap> GeneratorsOfGroup(g);
[ (2,7)(5,10), (1,3,5,7,9)(2,4,6,8,10), (2,10)(5,7) ]
gap> IsSolvable(g);
false
Since finding Galois groups is computationally hard, GAP also
provides probabilistic methods that return a list of the most
likely Galois Groups:
gap> f:=x^10-8*x^9+27*x^6-33*x^3+34*x-7;;
gap> ProbabilityShapes(f);
[ 45 ]
gap> TransitiveGroup(10, 45);
S10
HTH,
Nikos
> - Jordi G. H.