> -----Original Message-----
> From: maxima-bounces at math.utexas.edu
> [mailto:maxima-bounces at math.utexas.edu] On Behalf Of Jordi
> Guti?rrez Hermoso
> Sent: Thursday, January 24, 2008 2:53 PM
> To: Hugo Coolens
> Cc: maxima list
> Subject: Re: [Maxima] solving sextic by radicals
>
> 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.
>
> - Jordi G. H.
If I were doing it I would use the Algebraic Geometry tools in Maxima (or other programs) to convert the polynomials to triangular system
f0(x0)=0,f1(x0,1)=0,f2(x2,x1,x0)=0 .. etc
I just looked at the wikipeda article on Groebner basis:
http://en.wikipedia.org/wiki/Gr%C3%B6bner_basis
I can't do any better; in particular look at "Solving equations" and "Conversion of parametric equations".
Be aware that certain situations (double zeros/duplicated varieties I think) cause Maxima, Maple, and Axiom to wander off into no return land. I hope to find out why when I retire (hopefully shortly). This is personal experience and I am not willing to defend it right now.
RayR