I am an undergrad senior math major taking a gap year looking to become an actuary. However, I still want to continue learning pure math. I've been looking for a relatively high level text to self study for the next few months. I'm already a good chunk through the first chapter of Lang's "Algebra" and everything is flowing nicely. However, I have not done any of the exercises. Also, there are a lot of topological examples and things not purely algebraic which I would need to review (I have access to Munkres). In the end, I want to learn a lot of math (which could possibly help with my thesis) and solve a lot of problems. For background, my algebra class sophomore year used Artin's Algebra and I have also taken a seminar on algebraic combinatorics. Are there other texts (within or outside abstract algebra) better-suited for what I'm looking for? And finally, is the Companion to Lang's a supplement/fleshing out of the material, or more of a guide to getting at the solutions?



Lang's "Algebra" as a self-study book
In my opinion, Lang's *Algebra* has an excellent choice of topics for someone who wants to do further work in algebraic number theory or algebraic geometry. I'm not sure whether I'd recommend it for self-study, however. My feeling is that the exposition, and the exercises, are rather uneven. If you're studying from it on your own, I think you could easily get stuck at various points where it's Lang's fault and not yours. But that's just my experience, and your mileage may vary.


Since you mentioned algebraic combinatorics, I feel obliged to mention Richard Stanley's *Enumerative Combinatorics*. This contains a ton of excellent exercises with estimated difficulty ratings and solutions, so I think it's perfect for self-study. Algebraic combinatorics is my field, so I'm biased, but I'd recommend Stanley over Lang for your purposes.


