Janos Blazi asks:
> I should expect that Maxima (or for that matter, any CAS)
> should factor in the algebraic closure of Q and Q[i].
> But Maxima does not...
Maxima does solve 2,3,4-degree polynomials (and their functional
compositions using polydecomp, try x^6+2*x^4-x^3+x^2-x-1) using radicals
-- in the Solve command. Although *some* polynomials of higher degree
(e.g. x^5+20*x+32) can be solved in radicals, Maxima does not support
that. Check out the structure of factor(x^5+20*x+32,q^5+20*q+32),
though.
As for the Factor command, it is not intended to find roots, only to
factor over the rationals and (using the second argument) over specific
algebraic extensions. You could, of course, easily write a factor
routine that uses Solve to factor general polynomials up to degree 4
(and their functional compositions, of course). I am not sure how much
insight one gains from these messy radical solutions, though --
especially since they are not necessarily in the simplest form, and
simplifying them is very difficult.
As for factoring polynomials in radicals in general, I believe there has
been some progress in practical algorithms for this, but frankly I have
not followed that field. Do other CAS's support it? Is it useful?
Would you be willing to implement it for Maxima?
-s