7 Yes. There is a description of (a slow) one in van der Waerden even. If you are interested in implementations, Pari/GP, Sage and Magma will do it if the degree is not too large. Felipe Voloch Apr 29 '10
3 GAP also computes Galois groups, and it even finds explicit formulas for the roots (this makes for very, very impressive formulas!) when they can be gotten by radicals---you need to install the Radiroot package. Mariano Suárez-Alvarez♦ Apr 29 '10
7 The comments and SJR's answer show that there are indeed algorithms to compute this. But all of these suggestions are so far from effective, they can only be considered 'existence proofs' of an algorithm.
This is in fact a very active area of research, although it seems that most of this work has fallen completely under the radar of mainstream mathematicians, but this has been kept alive by a rogue band of mathematicians often calling a computer science department their home. Enough polemic, on to actual results. I find Alexander Hulpke's Techniques for the Computation of Galois Groups especially enlightening. Certain subcases, like that of the symmetric and alternating groups, can be found even more quickly (see Fast recognition of alternating and symmetric Galois groups ).
Even better, there are excellent implementations of recent such algorithms in GAP. Thus these computations are doubly effective. answered Apr 29 '10
