TITLE: Software for Galois Theory
QUESTION [15 upvotes]: 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 ?

REPLY [4 votes]: I haven't checked it out myself, but have you looked at "Computing Galois Groups with Mathematica" in the Wolfram Library Archive? From the description, it may do what you want:

Mathematica can be used to compute and form Cayley tables of the Galois groups of polynomials in Q. In addition, Mathematica can actually define a field extension and directly produce the elements of the Galois Group.