Calculating Galois Groups in Mathematica

If you are ( like me ):
- ( relatively ) new to Galois Theory
- looking for software to support your study of Galois Theory
- prefer software written in Mathematica because you know your way around in it,
then you should continue reading this post.

This year I became quite a fan of the sites of StackExchange. It is a priceless source of readily available know-how. I did not know where to begin looking for Galois Theory Software ( although I knew it existed ) so I posted a question in Mathematics StackExchange here: http://math.stackexchange.com/questions/93689/software-for-galois-theory As you can see I got answers fairly quick. It seemed that Sage, Magma have built-in support for Galois Theory. Both Mathematica and GAP have add-on package solutions. Needless to say these solutions will differ in capabilities, speed and so forth. But I wanted to focus on studying Galois Theorym and not wander off in software land. I accepted the Mathematica answer and pursued that route.

The package did not work!

Written more than a decade ( make that a century or more in software time ) ago or FIVE major releases of Mathematica ago. It got terminally deprecated. Function names used in the package were used in later releases of Mathematica with other, new functions. Other used functions got deprecated and were finally terminated. Software written with an older release can only be opened through the compatibility manager in Mathematica which is quite good at fixing issues. Not this time, which I ascribe to the sheer age of the package. In my confusion I posted the following question in Stack Overflow : http://stackoverflow.com/questions/8624000/how-to-handle-tag-arrow-is-protected-message-in-mathematica

With some help I was able to correct the issues. I don't know how or where to post it because the download came from here: http://library.wolfram.com/infocenter/Articles/2872/

So if you don't want to go the same issues just follow the posts above. I am of course willing to share my version 8 compatible version of the package.

Books on Galois Theory

Starting a self-study project involves finding the right books. I found the following books on Galois Theory which are aimed at beginners in the topic. For my purpose the following books are the most useful.

Galois Theory by David Cox, Wiley 2004
This is a very beautiful book of close to 600 pages, every page shows that the author loves the subject and really tries to explain the subject. It has sections on Galois Theory in Mathematica. ( In study-tech terminology: it effectively handles the first barrier to study, i.e. lack of mass, by making the subject tangible in the form of Mathematica functions. The student can explore the subject in a concrete fashion. ) Essential for self-study it has hints to selected exercises.

Exploratory Galois Theory by John Swallow, Cambridge University Press 2004
The author wrote software of his own in the Mathematica language which is available for download.

Galois Theory (3rd ed) by Ian Stewart, Chapman & Hall, 2004
Ian Stewart (*) is well know in England ( and beyond ) and this was his first book. I have added this book to my list because there are four different proofs of the Fundamental Theorem of Galois Theory in it.

Most good introductory books on Abstract Algebra have a few chapters on Fields and Galois Theory but it seems they are merely included to create an appetite for more.

Choosing books is a critical phase in self-study. I started with the book of Weintraub which he probably started very enthusiastically but it got denser and denser almost by page. ( Good book but needs extensive lecture notes by a teacher. ) Via Google Books -and other sources- you can find and browse any book on the subject. The reviews on Amazon can be helpful too.

The books are decided upon. On to phase 2: planning.

Galois Theory (3)

Equations of degree 5 and higher can't be solved. That's what I thought. I knew that certain equations of degree 5 could be solved however like $x^5 - a = 0$. Abel's impossibility theorem ( aka Abel-Ruffini theorem ) says that the quintic can't be solved in general by radicals. In general. What they in fact proved is that some equations can't be solved by radicals. Galois developed a technique for determining if an equation can be solved by radicals or not. This technique was the start of Group- and Galois Theory. A polynomial has a corresponding group, it's so called Galois Group. If this group is solvable, i.e. is a solvable group then the equation can be solved by radicals.

Galois Theory (2)

The polynomial F(x) = x^8 - 40x^6 + 352x^4 +960x^2 - 576 has 8 ( real ) roots : +/- Sqrt(2) +/- Sqrt(3) +/- Sqrt(5). ( See image above. ) The symmetry group of the roots is abelian and has order 8, all elements have order 2, so the group is isomorphic to C2 X C2 X C2.

I have found another introductory book on Galois Theory for undergraduates ( to support my study of Weintraub's Galois Theory I think, for the moment anyway. )


It comes with AlgFields ( download here ), a system of Maple and Mathematica routines for calculating with low-dimensional number fields and finite fields.

Galois Theory

While I was procrastinating on revising M208 stuff I explored new mathematical territory: Galois Theory. What have I discovered sofar?

- In between the fields Q and R there is another ( perhaps hypothetical field ) called A, the field of algebraic numbers. It contains of all quotients plus all numbers that are solutions to polynomial equations with coefficients in Q. For example Sqrt(2) is not a quotient but can be expressed as the solution of the equation x^2-2=0. So A is equal to Q plus all numbers like Sqrt(2).

- Something very interesting happens if we add ( adjoin ) Sqrt(2) to Q: Q remains a field! ( The field Q is an abelian group for + and *, the operations + and * are related via the distributive laws. Identities are 0 for + and 1 for * ). It can be proved trivially that { x | x = a + b*Sqrt(2) , a,b in Q } is a field. This field is written like Q(Srt(2)), or Q/(X^2-2) and is called an extension field.

- If we put on our Linear Algebra glasses we could say that a + b*Sqrt(2) is in fact a vector (a,b) over the basis {1, Sqrt(2)}.

- The roots of the equation X^2-2 have a C2 symmetry, the roots of X^3-2 have a Dihedral Group 3 symmetry. Investigating the symmetry of the roots of equations is a task in Galois Theory. The symmetry group is called the Galois Group, Gal(E/F). In our example E=Q(Sqrt(2)) and F=Q.

- Now the Fundamental Theory of Galois Theory ( FTGT ) says that there is a 1-to-1 correspondence between subgroups of Gal(E/F) and fields intermediate E and F.

Fascinating stuff. Unfortunately Galois Theory is not part of any Open University course I know of.

The book I am reading on Galois Theory is:

Galois Theory

Galois Theory is a topic which is, at least in the algebra books I have, covered in the last chapter as the most beautiful result of algebra. I know that Galois introduced group theory and proved that it was impossible to solve an equation of type f(x)=0, where f(x) has a term of x in the 5th degree or higher, by means of  a formula. ( Solving the quintic by radicals is how it is described. ) What bothers me is that I still can't follow the proof, or worse: I simply don't get it.

I found a hint though. The Galois Group of x^2-1=0 is C2 and of x^4-2=0 the Galois Group is the Dihedral Group of order 8 ( symmetry group of the square ). Will play a bit with these examples, I hope it will break some ice.

Update: the field we work in is Q.

Abstract Algebra and Mathematica

I found a promising e-book on ( Explatory ) Galois Theory. The book is written at the undergraduate level and it has 'Explatory' in the title because the text uses and refers to the Mathematica ( and Maple ) packages.

What is Galois Theory anyway? Galois only wrote one ( unfinished ) paper before he died at the age of 19 in a duel, but the contribution he made to mathematics is very significant. Before Galois mathematicians were searching for a formula to find the roots of a fifth degree polynomial equation.



Galois proved that it is impossible to find such a formula. It does not exist. In his proof he introduced the concept of a Group.

Abstract Algebra and computers. I learned a lot of concepts in Abstract Algebra by using the package of Hibbard and Levasseur: Exploring Abstract Algebra with Mathematica. It is a book and a software package combined as one. The package though, can be downloaded freely. Mathematica (6.0) is a requirement as the package is written in the Mathematica language.