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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 ?

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3 Answers 3

up vote 3 down vote accepted

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.

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I hoped to find a solution for my favorite software Mathematica. Although I am sure the others can do it as well ( or better, who knows ). –  ndroock1 Dec 24 '11 at 8:54

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.

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The example returns an error >> GaloisGroup(x^6+3); ^ User error: Identifier 'x' has not been declared or assigned but I can have a look at the Magma manual later. Thanks anyway. –  ndroock1 Dec 23 '11 at 15:56
    
@ndroock1 I am sorry. The angle brackets got swallowed up by markup. Now corrected. The point is that you need to define x as a free variable (that's what the first line is supposed to do). –  Alex B. Dec 23 '11 at 16:14
    
In PARI/GP: polgalois(x^6+3) –  Charles Aug 26 '13 at 17:17

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

See this page for an example.

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Haven't looked at Sage for a long time because I invested a lot of time in learning Mathematica, but I suppose it's time to give it another look. –  ndroock1 Dec 23 '11 at 15:53

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