Computing Galois groups (in response to sen1)

I believe that the package SYM has some of the functions
that you need for computing Galois groups. 
The documentation of this package (in French) says that it can perform
some manipulations with elementary symmetric polynomials that may be
useful for Galois group computations. As I don't know much about 
Galois theory, I don't fully understand this documentation. 

I think you can find more information on the applications of this
package by searching for Valibouze (the author of this package) 
on Citeseer (citeseer.ist.psu.edu/). Annick Valibouze has also 
a web page at the university Paris 6 http://www-calfor.lip6.fr/~avb/
where she describes some of her work on Galois theory.

hope that helps.