# 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.

• 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. 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). Dec 23 '11 at 16:14
• In PARI/GP: polgalois(x^6+3) Aug 26 '13 at 17:17
• This method is very useful.For example, (pari.math.u-bordeaux.fr/gp.html) (polgalois(x^5+5*x+1)) or (magma.maths.usyd.edu.au/calc) (P<x>:=PolynomialRing(Rationals());GaloisGroup(x^5+5*x+1);) Dec 5 '19 at 6:54

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