Software for Galois Theory Background: While studying Group Theory ( Open University M208 ) I had a lot of benefit from the Mathematica Add-on package AbstractAlgebra and later from the GAP software. I am currently self-studying Galois Theory ( using Ian Stewart's Galois Theory  ). 
Question: Is there a program that calculates the Field Extensions / Galois Group for a ( simple ) polynomial ?

Canonical answers are Sage, Pari, Magma. The first two are open source, the last one costs money but has an online calculator. Type for example
P<x>:=PolynomialRing(Rationals());
GaloisGroup(x^6+3);
in the online calculator and hit submit. See the online manual on how to interpret the result.
