Self teaching Galois Theory

At uni, I did a module in group theory which I really enjoyed. I also did one on number theory which had aspects of rings and fields in it and I enjoyed learning this. Now that I have finished uni, I was thinking that I would still like to learn all this stuff for two main reasons: $1)$ I like it and it was interesting and it's a fun "hobby" to have, and $2)$ If I wanted to apply for a masters in the future, it'd be good to show them that I have been doing some maths in my spare time.

I spoke to a friend and told him I liked abstract algebra and in particular, I liked group theory. He said that if I liked that then Galois theory might be a good subject to look at, but I am a bit worried about going this advanced without knowing if I completely understand the basics. For example, I can create semi direct products for cyclic groups, but not really any other groups. I don't really get how to use cosets, etc. So I was thinking, would it be a good idea for me to first start on a basic book like "A First course in Abstract Algebra" as although there will be bits I understand quickly, there will be bits I have forgotten/didn't learn properly, etc.

However, my friend said that it might be better to learn Galois theory with a good book as any basic bits I will remember again or if I genuinely don't know/remember, then I can look it up later.

What would you think is the best idea?

Apart from Galois theory, are there any other interesting subjects within the field of Group theory I could do?

• Learning Galois theory sounds like an excellent idea. You could learn some representation theory and/or Lie theory, though those might be more difficult. Algebraic topology makes use of a lot of group theory, so that could also be worth looking at. Jun 12 '13 at 19:18
• @Julien I did a module that I really liked which was called "Analytic methods in higher geometry" and in there we did things like symplectic structures, wedge products, tensor products, etc and I enjoyed that module. Are there any "further" aspects of maths for these topics that I could do? Jun 12 '13 at 19:25
• I'm not terribly familiar with those topics, but I imagine you could find plenty of new material to learn about symplectic geometry (or just differential geometry in general). Jun 12 '13 at 19:54

A good book to self study from, since it contains solutions to all the problems, is Abstract Algebra: The Basic Graduate Year by Robert Ash, available for free here. You can read about the style in which he writes his books here.

• Weren't you Barack Obama some time ago? Jun 13 '13 at 0:59

I would jump straight in at the deep end. Try Stewart. I haven't read his Galois theory book, but his other books are very accessible, so you should manage on your own. You can Google online lecture notes for pretty much any background material you need.

I recommend A Course in Galois Theory by D. J. H. Garling. This was the textbook I used and I highly recommend it. At the time I was taking group theory concurrently, and had no previous experience with it, and I did just fine. You will be more concerned with field theory.

I recommend Galois Theory by David Cox. But, if you want to learn more theory you can read book of abstract algebra, like Rotman, Dummit, Ash (these books have a lot of theory about field extensions). Nowadays, I'm developing my thesis about Kronecker - Weber theorem and I'm using these books.

As a person in the same positon as you, I am finding Serge Lang's Algebra an indispensable learning resource.

It can safely be read non-linearly. So right now, I've started taking notes from chapter V Algebraic Extensions, which covers all the required field theory for Galois theory.

Chapter VI Galois Theory covers even more than an average book on the topic, and it includes a section called "The modular connection" which gets into some deep mathematics.

There is also a Galois cohomology section, which is very cool. After learning field and Galois theory, it shows you how to apply it to a very interesting type of cohomology theory.

Chapter VII Extensions of Rings is essentially goes over the first steps taken in any Algebraic Number Theory book.

Not only that, this book goes all the way to General Homological Algebra and covers group cohomology in the exercises there.

It is a also packed full of exercises.