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]
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.
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.
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).
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.