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Thursday, October 14, 2010

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:


  1. I think you will find Galois theory is covered in Coding Theory M836 (Postgraduate course). I tried to get the O.U to let me do that module as a standalone course but they said no.

  2. Thanks for your reply!

    That's good news, although it will take a while before I qualify for doing post-graduate courses.

    The names of the OU mathematics courses don't do justice to its contents. I.e. 'Pure Mathematics' = Linear Algebra + Group Theory + Real Analysis.

    The course names are a bit cult-ish. I mean if you do a Linear Algebra course at university A in country B then it is probably called MXXX - 'Linear Algebra' with a predictable contents. In the group of OU math students everyone knows the contents of MST121. Outside the OU one can only guess what is meant by a course called 'Using Mathematics'.

  3. I agree with you. I have an interest in digital signal processing and am doing O.U maths to get my maths back to a level where I can understand the DSP text books. Figuring out which O.U maths modules to study is a science in itself.

  4. This post gives me bad memories of my maths degree at Leeds Uni. Galois Theory was one of my year 3 pure modules, which was supposedly the toughest module of the toughest course at Leeds Uni in 2002. I found it fairly straight forward and interesting personally. I can't remember much about it other than it was to do with solving n'th degree polynomials.

  5. Sorry for the bad memories ;-) I hope you found an antidote somewhere else on the site.


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