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. – hasnohat 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? – Kaish 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). – hasnohat Jun 12 '13 at 19:54