Guides/tutorials to learn abstract algebra?

Solution 1:

I can highly recommend "A Book of Abstract Algebra", by Charles C. Pinter. You'll learn about groups, rings and fields. You will also learn enough Galois Theory to understand why polynomials of degree higher than $4$ are, in general, not solvable by radicals.

It is 'formal' in the sense that it is rigorous, but the author is also very good at explaining the intuition behind all ideas. It is much less dense than most Abstract Algebra books, and, in my opinion, and excellent introduction to the subject.

Furthermore, it isn't expensive and it contains solutions to numerous exercises. See the amazon page of the this book for more positive reviews. Again, highly recommended!

Added: once you finished this book, you're ready for more advanced treatments of abstract algebra. After Pinter's book, you could try "A First Course in Abstract Algebra" by John B. Fraleigh. After that one, a great option is "Abstract Algebra" by Dummit and Foote. This is quite an advanced textbook, but a good one nevertheless. Once you've worked your way through these books (I advise you not to just read through them, but actually soak up the information by doing the exercises and reading actively) you will have a strong basis of knowledge in abstract algebra. By then you can tackle more advanced topics.

Solution 2:

Great video lecture series by Benedict Gross at Harvard:

http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra

Outstanding short pieces on many facets of algebra by Keith Conrad at UConn. Very clear exposition and lots of intuitive discussion any plenty of warnings - teaching at its best(even one on Rubik"s cube):

http://www.math.uconn.edu/~kconrad/blurbs/

Solution 3:

"Abstract Algebra: Theory and Applications" could interest you. It is a free textbook and while it is not written informally, it is easier to understand than usual Abstract Algebra textbooks. I think it is a good way to get into Abstract Algebra.

http://abstract.ups.edu/

Solution 4:

Nathan Carter's Visual Group Theory.
Janet Chen's Group Theory and the Rubik's Cube. (Available for free but a little more formal)
Wildberger's Informal Introduction to Abstract Algebra.
This question may help you to find out on the unsolvability of polynomials of degrees higher than 4.
Avner Ash's Fearless Symmetry: Perhaps this may be of interest to you, at the end of the chapter 1, the author states: [...] our goal: mod p linear representations of Galois groups. We explain how these representations help to clarify the general problem of solving systems of polynomial equations with integer coefﬁcients, and how they can sometimes lead to deﬁnitive results in this area.

This is the most informal and written in a moderately comprehensive references I can remember - but you didn't tell us how informal or how comprehensible the references should be, If you can provide us a example book of what should be informal and comprehensible, I can add a little more to the list.

Solution 5:

I really think that Isaacs book Algebra: A Graduate Course introduces the group theory in detail without omitting any proof. It may sound difficult because of the adjective "Graduate" but I do not think that the explanations are that difficult to follow for undergraduates as long as they know how to write proofs.

The best freebies for algebra in my opinion is Milne's website (http://www.jmilne.org/math/). Not every note is complete, but his excellent notes tell you which books to buy to corroborate them.