Advanced Algebra textbooks

Eh, is this supposed to be a good book because I have no idea why the definitions aren't clearly marked and indexed. I only checked the chapter about Group Theory but I was not impressed. Maybe for a quick review of the topic it might be enough but for a beginner it seems that it is not rigorous. The definitions could be much more clear explicit. And there is no reason why they should not be indexed.

I don't like the typical definition-theorem-proof approach of most textbook in mathematics, including these. It's great for a classroom, no good for self-study. As an alternative, I highly recommend A Book of Abstract Algebra by Pinter. If you work through that first, you may actually enjoy these two later.

FYI, http://www.genealogy.ams.org/id.php?id=8127

Can't agree more. Definition-theorem-proof type of textbooks is way too clean. They don't tell you how ideas came to be or why they mattered. In other words, it's hard for students to learn the intuitions behind the ideas. I wish there are list of "XXX from Ground-Up" type of books that show readers a list of problems, struggles of people trying to solve them, and how ideas emerge from the numerous attempts. Leslie's paper Paxos Made Simple was written in that way. A few chapters of Kleinberg's Algorithm Design were written in that way too.

The best undergrad algebra textbook I've studied is Algebra by Mac Lane and Birkhoff (3rd edition! the previous editions aren't quite as good and substantially different; i haven't seen the 4th edition and it is out of print so /shrug). I've used multiple books both in self-study and class and this book is, to me, in a league of its own. Not only does Algebra teach modern algebra, it teaches one to think like a modern algebraist, and not like just any modern algebraist, but like Saunders Mac Lane who was pretty great at algebra.

As an example of Algebra's approach, take the isomorphism theorems [1]. Now many undergraduate textbooks (like Dummit and Foote) will prove these theorems by manipulating cosets and deal with gross "implementation details" at the level of sets. Mac Lane insists otherwise: The only time you have to manipulate cosets is in order to construct the quotient G/N of a group G by one of its normal subgroups N. Once you have constructed this group and proved its universal property, the isomorphism theorems can be proved without ever mentioning cosets again. What is that universal property? It has two parts: First is that there is a morphism p from G to G/N which sends all of N to the identity in G/N. Second is that any morphism f from G to any group L that sends all of N to the identity in L necessarily factors uniquely up to isomorphism as a composition of morphisms g ∘ p. This is the essence of a quotient group.

Mac Lane's approach is to apprehend the essence of what is studied while discarding as much of the set theoretic husk as is possible. It is algebra in its purest form, accessible to and transformative of the mind of an undergraduate. Reading this book is a recurring joy to me.

[1] https://en.wikipedia.org/wiki/Isomorphism_theorem