Studying Mathematics Thread

[Kernel Sohcahtoa]

Thanks for your reply, Polymath.  Galois Theory sounds exciting; however, I'm eager to take a break from abstract algebra and explore another topic.  

That said, I've studied the following topics: groups; fundamental theorems of groups; cyclic groups; subgroups; direct products; functions; symmetric groups; equivalence relations and cosets (I really enjoyed this section); counting the elements of a finite group; normal subgroups and quotient groups (this section was neat); homomorphisms (these are cool); homomorphisms and normal subgroups (I enjoyed this section; it covers the isomorphism theorems, which are very neat IMO); Rings (I enjoyed this section).  I'm currently studying subrings, ideals, and quotient rings.  Once I'm finished with this section, I'm either going to pack it in and conclude my self-study or cover the section on ring homomorphisms and then call it quits.

You might try some basic algebraic topology: the fundamental group is readily accessible and leads to lots of interesting ideas.

Polymath, I want to thank you for mentioning Galois Theory and for mentioning the proof about quintics: it seems very interesting that for n is greater than or equal to 5, there is no general formula for finding the roots of nth degree polynomials in terms of radicals.  Getting underneath ideas like these, understanding/enjoying them, and gaining an appreciation for them, are the reasons why I chose to self-study mathematics as a hobby.

I will complete the chapter on ring homomorphisms.  Completing this section, along with the others that I've completed, will be  equivalent to a course in Abstract Algebra I.  Afterwards, I plan on studying the following topics:  polynomials; polynomial rings and fields; Unique Factorization Domains; Extensions of Fields; Normal and Separable Extensions; Galois Theory; Solvability.  Based on my understanding, this second stretch of material will be the equivalent of a course in Abstract Algebra II.    

P.S. You've motivated me to continue my studies of abstract algebra.  Thank you.