Is Aluffi's "Algebra: Chapter 0" a good introduction to algebra?
I am not extremely familiar with Aluffi's book, but I heard it is really good. As for your questions:
- Many fields of mathematics can be learned in more than one way. I think that the category-theoretical approach adopted by Aluffi is really nice, and teaches you not only algebra, but basics of category theory as well. This is good because if you ever decided to look in depth at categories you would already have a baggage of examples behind you.
- You can try and post some exercises here together with your solutions. I am sure that many people will be happy to help you understand if you are working properly.
- Aluffi treats Galois theory (chapter 7, section 6). He doesn't seem to be treating group representations, but you can always find some other reference to study the subject.
All in all, I think the book can be a really good place to learn algebra. Obviously this is only my personal opinion, there will certainly be others (probably knowing much more than myself on the subject) with different views on the subject.
I have some familiarity with the book and believe it is a good one, though there are others I like more.
Algebraic arguments are generally more elegant, and more enlightening, when one works with arrows rather than elements. Don't worry about the book not being "serious" enough; the whole point of the text is to start you off thinking with the same language as "serious" mathematicians in algebra-heavy disciplines. The main reason to stay away from Aluffi is that category theory is rather abstract, and can seem difficult and/or pointless until one has built up a library of examples. This doesn't seem to be a problem for you, so Aluffi is probably a pretty good choice.
Saal, I second the opinion, based on starting the book, that Aluffi has one of the most user-friendly intros to category theory around. And that is a very good thing because one can relatively easily make one's way through the basics of abstract algebra then hit the wall of categorical thinking and get lost and discouraged. And given its importance as well, it is very nice that you seem to be getting it.
As others as pointed out, if you want more on group representations, you can look at Dummit and Foote, or Lang, and I'm sure Jacobson's Basic Algebra (yes, it has long chapters on Galois Theory in vol. 1 and Rep. Theory in vol. 2). Also, Emil Artin's little book on Galois Theory I recall as being very concise and clear.
Indeed, if you want something that is a bit more 'concrete,' and less concerned with arrows and diagrams, then in increasing order of difficulty, these are excellent books. I am currently concentrating on Dummit and Foote while starting to look at Aluffi for decent insight:
Birkhoff and MacLane, Survey of Modern Algebra. 4th edition very nice. Undergrad level.
Herstein, Topics in Algebra. Intermediate in level between undergrad and grad. Beautifully written.
Dummit and Foote, Abstract Algebra. Excellent. Has some of the topics Aluffi leaves out.
I find Artin's Algebra and bit quirky, and he leaves out proof details I'd like to see sometimes. Lang's Algebra is also quirky, but interesting and encyclopedic.