Book recommendation for Abstract Algebra

Question

Hello fellow members of Math StackExchange!

Since this is my first post here, I should start with a little introduction of myself. I've recently finished the second year of math at university and I did fairly well in the classes I took this semester (multivariable calculus, intro differential equations, intro linear algebra, intro abstract algebra). Out of the math topics I've been exposed to, I found myself to be drawn more to discrete math (algebra being by far my favourite). For example, I loved the definition-theorem-proof pattern that the abstract algebra course this semester followed.

The algebra course at my school was quite standard, following the groups-rings-fields path. We used Abstract Algebra: Theory and Applications by Tom Judson. We did skip over some chapters in the book, but I will finish learning them in my own time because I'm taking the summer semester off. However, because I get four months off in the summer, I'm hoping to also learn from a more advanced textbook, which leads to the main question of the post:

What do you think is a good second textbook for self-studying abstract algebra for someone who has taken a semester of algebra?

I know algebra is a big topic, so I'm sorry for posing a potentially vague question. Also, I searched this question on the forum before and the posts that I came across seem to be for beginners, so hopefully this isn't a duplicate.

Best regards,

Kevin

Answer 1

I'll throw in my 2 cents.

Yes. Algebra is a huge field. If you are looking for survey book, Dummit and Foote's Abstract Algebra text is pretty hard to beat. It's written at a very reasonable level. It has detailed examples where other books don't bother.

Another text I've really enjoyed is Rotman's Advanced Modern Algebra. Rotman's text (like Dumitt and Foote) covers a bit of everything. Rotman starts with a lovely discussion about the beginnings of group theory motivated by the theory of solvability of polynomial equations. There's interesting bits of history splashed in here and there. He also tends to prove things several times - just to display different techniques (this I love).

One thing that Dummit and Foote lacks is much of any category theory. Rotman does quite a bit in this direction. One word of warning through: Rotman's book wanders around. This makes it a bad reference, but who cares - it's a fantastic read!

Answer 2

Paolo Aluffi's Algebra Chapter 0 is a good text that looks at things from a slightly different perspective as it introduces categories early on.

Another good text (it is not that advanced, but it does cover a lot of topics) is Richard Dummit and Richard Foote's book called Abstract Algebra. I think that this book is excelent.

Answer 3

This recommendation comes from my own experience of self-studying:

Get yourself equipped with two books and start studying them in parallel:

1. Introduction to Commutative Algebra by Atiyah and MacDonald.

2. Galois Theory by Emil Artin.

Both books are about 100 pages each and they are extremely well written and fun to read. And you would need to go through them at some point for sure if you are to choose a path in algebraic geometry or algebraic number theory in the future.

Answer 4

Topics in Algebra - Herstein is a great second book. It assumes no knowledge of algebra but it is tougher than your standard intro book. In my opinion, tougher than Judson. If this book is too "introductory" then I would recommend:

Finite Group Theory - Isaacs. The first chapter is Sylow Theory, so it definitely assumes you have adequate knowledge in algebra.