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?
Best Answer
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.