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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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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 ). – nilo de roock 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



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. – nilo de roock 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. – nilo de roock Dec 23 '11 at 15:53

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