Studying Mathematics Thread

[polymath257]

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.

Galois theory is actually why group theory got going. There is some wonderful structure here relating roots of polynomials, isomorphisms of fields into themselves, and the structure of groups.

In any case, the progression you give looks good.

if you have any questions, feel free to ask! BTW, what book(s) are you using?

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I've heard about that proof for quintics and higher. I've not seen it, but I'm blown away that it's possible to prove things like that.

Prove, not "provide evidence for", like wanky science! Who needs it, right guys?

Well, at least part of the issue is defining what needs to be proved. Making the question of 'solving polynomials' well-defined as opposed to something vaguely felt is non-trivial.

And, of course, that is before you can even attempt the proof. Finding out how the structure of particular groups (alternating groups) relates to solvability of polynomials by radicals is just way cool.