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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.

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.

## Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)