Studying Mathematics Thread

[Kernel Sohcahtoa]

I'm getting close to finishing up my abstract algebra refresher self-study.  I can't stress enough how important it is to be able to take a pen and paper to a proof of a main theorem that is presented in a chapter and then being able to put it into your own words and make sense of it via identifying the various definitions, theorems, corollaries, lemmas, and other mathematical ideas that were involved in successfully writing the proof (this process is referred to as self-explanation).  Hence, IMO, once one gains a general understanding of proofs, then learning a proof-based math subject is a good way to hone one's self-explanation proof skills.

That said, once I'm finished, I'm debating whether to take a break from math or to jump right into teaching myself basic graph theory; IMO, graph theory seems like an interesting topic.

How far into the abstract algebra did you get? I always really enjoyed Galois theory. That there is a *proof* that 5th degree polynomials can't be solved via radicals is just *fun*. But you need to do quotient rings and some field theory first.

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.