Galois Theory (M3,4,5P11)

NOTE THAT THIS IS THE WEB PAGE OF AN OLD COURSE WHICH FINISHED IN 2015.

Example sheets.

This year's problem sheets and solutions:

Sheet one and solutions.
Sheet two and solutions.
Sheet three and solutions.
Sheet four and solutions.
Sheet five and solutions.
Sheet six and solutions.

[there are six sheets]

Hand-outs.

I stated without proof in the course that any field could be embedded into an algebraically closed field. Here's the proof.

We talked a bit about finite fields, but here's a proof that given a prime power p^n, there's a unique (up to isomorphism) finite field with this order.

A nice example of Galois theory is given by the theory of cyclotomic fields and cyclotomic extensions. I wrote some notes on this stuff here. You might want to work through it because it's a nice application of the stuff we learnt in the course so it's good for revision purposes.

Tests.

There were two tests, on 2nd Nov and 30th Nov 2015. Here are the questions and solutions.
First one (2/11/15) questions and solutions.
Second one (30/11/15) questions and solutions.

Mastery/Comprehension (M4,M5).

This is not for 3rd year undergraduates; this is the extra question that MSc and MSci students take.

Here is the mastery handout. It's on how the fundamental theorem of Galois theory works for infinite extensions. Here is also an extremely long example sheet and solutions. Please read the bit at the top which says that you don't have to do all these questions, before you have a nervous breakdown.

Times and dates.

The course ran from October to December 2015.

What's in the course?

I will talk about an amazing invariant that one can attach to a polynomial with rational coefficients, namely the "Galois group of its splitting field". It will involve a lot of work to even give the definitions we need to understand this idea. We'll start off with some abstract field theory, and we'll see what happens to a field if you make it a bit bigger by adding in a root of a polynomial that didn't have a root before (exactly like how you get the complex numbers by starting with the reals and then throwing in a solution to x^2+1=0). Already this will be enough to prove that you can't duplicate the cube, trisect an angle with ruler and compasses, or (assuming pi is transcendental) square the circle. Once we go deeper into the theory we'll even find out why there is no formula which just uses +-*/ and n'th roots and which spits out the roots of a general quintic (degree 5) equation: this is all to do with the fact that the symmetric group on 5 letters is (in a precise sense) far more difficult to understand than the symmetric group on 4 letters!

I gave this course in 2013 and it was OK. I gave it in 2014 and it was better, but I was still not happy with it. My goal this year (2015) is to actually give a good Galois theory course, with some nice proofs, not too much waffle, lots of examples, and then the killer theorem about the quintic at the end.

In 2014 someone took notes for the lectures! The lectures will not be exactly the same in 2015 -- I am still not 100 percent happy with the order of the material -- but I am sure the notes will be helpful. If anyone wants to type up the 2015 notes then feel free -- and I'd love to have a copy (and I'll read it through and tell you about all the mistakes I find, if you like).

Recommended texts.

My illustrious predecessor Martin Liebeck used to recommend these:

C. Hadlock, Field theory and its classical problems
I. Stewart, Galois Theory
J. Rotman, Galois Theory
J. Fraleigh, A first course in abstract algebra
I. Herstein, Topics in algebra

I personally bought this book when I was an undergraduate:

D. J. H. Garling, A course in Galois Theory

but actually I learnt Galois theory from my lecture notes, so perhaps I'm not the best person to ask about books.