Good abstract algebra books for self study

Solution 1:

There's always the classic Abstract Algebra by Dummit and Foote. Section II of the text gives a nice treatment of ring theory, certainly providing plenty of review for what you have already covered while introducing more advanced concepts of ring theory. Section III will cover the field and Galois theory you're interested in. Some of the exercises can be difficult at times, especially for self-study, but the authors tend to give a number of examples and always provide the motivation for why they are doing what they are doing.

Solution 2:

Fraleigh's "A First Course in Abstract Algebra, 7th Edition" is a good book for self study. It is easy and good for the beginners, and it has a complete solution manual written by the author.

Solution 3:

Try Contemporary Abstract Algebra. This one, I think, has lots of nice examples. The following is from Googlebooks:

"Contemporary Abstract Algebra 7/e provides a solid introduction to the traditional topics in abstract algebra while conveying to students that it is a contemporary subject used daily by working mathematicians, computer scientists, physicists, and chemists. The text includes numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings giving the subject a current feel which makes the content interesting and relevant for students."

Also, I would like to suggest you read this article in wikipedia. You may find the references valuable.

Solution 4:

Note: This answer is copied over from an answer I gave on a previous very-similar question, because it still applies here.

This is likely not going to be a popular suggestion, since it's relatively unknown, but I think the perfect book for you is Allan Clark's Elements of Abstract Algebra.

It's a unique book that covers the basics of group theory, ring theory, and even a tiny bit of Galois Theory, but it does it almost entirely through problems. Every chapter begins with a short section defining some terms and giving a few basic proofs, and then it leads the reader through the rest of the exposition in a series of problems, some difficult, some not. The end result is that if you actually do all the problems, you've written the book yourself. It's impossible not to be comfortable with basic abstract algebra if you take this book seriously.

It's also probably the cheapest book on this entire list :)