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Abstract Algebra suggestions

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Do NOT start with dummit and foote unless your mathematical maturity is at a strong level.
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I also do some self-study. I have read Pinter's book, and I liked it. I think it is a fantastic introduction to Abstract Algebra. I am currently doing Harvard's extension course on Abstract Algebra -which you can find on youtube- and uses Artin. I can say I am not a big fan of it. Someone else recommended Aluffi's chapter 0, which I have also tried (not the whole thing, of course), and I like it.

Edit: To elaborate on why I dislike Artin, I think the text is ok, and the book's content is quite appealing. I have read almost only the assigned readings for the course, but I checked the ToC, and it includes chapters on representation theory and sections on algebraic geometry. HOWEVER, I feel the exercises are not good relative to the text (1st ed.). Frequently I couldn't answer several of the assigned exercises because what was written in the text was simply not enough to answer them. I mean, it is ok to do some research for some questions, but I think the problem is quite generalized. This is a big killer for self-study.
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I learned algebra by self-studying Dummit and Foote and Artin together.

Lots of people are advising against D&F, which I get. It's probably better as a reference than a textbook, especially for self-study.

Artin's book, on the other hand, is a real gem. He conveys the flavor of the subject much better than most texts. A big part of this is emphasizing the ubiquity of linear algebra. It doesn't read like a modern textbook, which in my opinion is a good thing. It's more like Rudin's analysis books - terse but extremely rewarding with effort.
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Gallian is beautifully written. He really makes abstract algebra fun by mixing in fun applications and history.

I would advise against Dummit and Foote or Artin. Dummit and Foote is reserved for when you have a strong foundation in algebra.
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I really like Fraleigh as a first course text (admittedly I’m biased because that’s what I used.)

I largely agree with the other commenter that while I’ve read D&F, I’ve used it more as an reference or refresher than anything else.

Beyond that I can’t really comment on Artin’s book. Worth noting that I’ve heard a lot of good things about Joseph Gallian’s book as well.

All of these books have a sufficient selection of exercises to work through—which is obviously highly recommended for working towards understanding!
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Aluffi's new undergrad book: "Algebra: Notes from the underground" is really nice. I'm using it to self teach.

Edit: saw Aluffi's Chapter 0 recommended. It's probably not the best place to start. Maybe read it after Notes from the underground
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The undergrad Hungerford is a very gentle introduction to algebra. I think it's a very good option for self study.
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I wholeheartedly recommend Fraleigh's book, especially when it comes to group and ring theory.
The book is slow paced and has big number of examples. On top of that it has wide variety of exercises, from trivial to challenging.

The book will help you build strong foundation and intuition in abstract algebra and will allow you to learn some more advanced topics.

If I remember correctly, the book lacks some topics such as module theory, so having a book like Dummit-Foote as reference might be a good idea.

But since you are self-studying the topic, I would say go for Fraleigh.
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Dummit and Foot's is definitely not the first book you should read. I got through the first chapter in about a month and ny brain was fried. Switched to Fraleigh's and it was a breeze. It really makes a difference what the book's target audience is.
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I reccommend Joseph Gallian's book, but Charles Pinter's book is also pretty good but does not delve as deeply into the topics as Gallian, so Pinters book can be used as an introduction to the subject. I also enjoyed 'A concrete approach to Abstract Algebra' by W.W. Sawyer as an intro to the subject.

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