TITLE: Self-Studying Abstract Algebra; Aye or Nay?
QUESTION [26 upvotes]: I am a high schooler with a deep interest in mathematics, which is why I have self-studied Linear Algebra and have begun my self-study in Differential Equations. As I am a man who likes to plan ahead, I'm pondering what field of mathematics to plunge into once I've finished DE's. I am thinking of Abstract Algebra: it has always sounded mystical and intruiging to me for some reason. I have a couple of questions regarding AA:

What exactly is Abstract Algebra? What does it study? Please use your own definition, no wikipedia definition please.
What are its applications? Does it have a use for example in physics or chemistry, or is it as abstract as its name suggests?
Would it be a logical step for a high schooler to self-study abstract algebra after studying LA and DE's, or is there a field of (post-high school) math 'better' or more useful to study prior to abstract algebra?
What are some good books, pdfs, open courseware etc. on abstract algebra? links and names please.

REPLY [4 votes]: Here is a link to Harvard Math 122 (Extension 222) taught by one of the greats - Benedict Gross.
It offers a full series of videos along with complete notes taken by a very competent GSI which makes for a very thorough presentation of the first algebra course.
http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra
It follows "Artin" which I think is an excellent entry point for intuitive understanding. (An echo of the above remark regarding Dummit and Foote - which is much better when you get further in rings, modules, etc.) Should you choose to go this route and buy the text, I would recommend the more recent 2nd edition.
While I'm no expert on the math curriculum, most programs entail analysis, topology, and algebra. If you haven't yet undertaken real analysis, here is a link to a free downloadable beautifully transcribed lecture course by Fields Medal winner Vaughan Jones. This too I assure you is outstanding, and may be you best next step.
https://sites.google.com/site/math104sp2011/lecture-notes
Lastly, not knowing what you used to study linear algebra, you might want to take a look at Axler's "Linear Algebra Done Right." I would do that after the real analysis and before the algebra material.
P.S. Since these materials are available at no cost, you might want to just take a taste- like chocolate, eating is believing.