Software for Galois Theory

Question

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 ?

Answer 1

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.

Answer 2

SAGE can calculate the Galois groups of number fields (as a group of permutations).

See this page for an example.

Answer 3

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.