Algebra

Rating:
Summary: Good overall, not consice enough
Review: Artin is a good intro book for casual self-study in a lazy summer afternoon. That is, if you are bored and decide to learn some abstract algebra just for fun, go for Artin. It's very easy to read and the author basically does the thinking you have to do for you in the text. However, if you want to be hardcore and be more involved with the material, I would recommend Dummit and Foote, a more rigorous treatment.

Rating:
Summary: Great book for challanging you to think with clarity
Review: Artin's book is probably one of the better books, more because of the way you have to read it to learn it. Artin's book is extremely nonstandard, in the sense that it isn't so "encyclopedic" as you usually encounter with the whole theorem, corollary, proof, proof, proof, example, example sequence. What I think a lot of readers miss is that Artin's book makes you fill in the details he leaves out by using the hints he mentions in words within the text. For example, I was able to expand the two pages of notes on Ch 2, section 5, in Artin into about 8 pages of original notes and theorems, just by digging for the main points. If you want a sample of my notes, please email me and I'll email you a brief PDF sample for you to compare. That being said, assume that you will have to dig a lot in this book, and should you choose to study from it, I suggest the following:

How to read it:

With a cup of coffee, or tea, and a notepad of paper for you to make comments on. Do not take notes; anyone knows that simply rewriting things doesn't do anything for learning. You should do the proofs in different ways, if you can see how, and try to make some of the aside remarks he makes into theorems or more precise ideas (this is not to say that Artin lacks rigor; this is just talking about the general commentary. When he makes commentary, it always seems to be enough to actually dig out exactly what to do after a little scratching). He also leaves a lot of easier proofs to the reader, so do them.

Is non-standard a less-rigorous approach?

No. Artin is definitely doing his own thing here, but I think it works really well. Getting through that book FORCES you to take responsibility for your math education by making you get your hands dirty while also developing an intuitive understanding of algebra.

What about his personal flavor of algebra?

Well, it's fairly clear to all of us that texts seem to have different flavors (being a function of the author's research area, and what was fashionable during the time the book was authored). Artin's book is algebra with light strong hints of geometry throughout, as he is in algebraic geometry. You will find that unlike most authors, Artin loves structures made of matrices when working with examples, as opposed to permutation groups or the ``symmetries of the square group,'' known also as the ``octic group.'' While these things have their place in his book, he changes the emphasis here. That's why I suggest using a companion book so as to have two sharply contrasting flavors of presentation, and Herstein seems to write in such a way that would do this. Artin covers a lot of material extremely quickly, but focuses on the bigger picture in several key areas. For example, the sections 7 and 8 in chapter 2 deal almost exclusively with how one would go about investigating a particular group structure to learn about it, teaching a student how to dig into something they might barely understand.

Advice to make a wondeful course:

Use another book which IS encyclopedic as a reference, since Artin doesn't label theorems and definitions so explicitly. I suggest Herstein's Abstract Algebra, or his book Topics in Algebra.

Personal Charracterization:

I place this book as one of my favorites on the bookshelf, and it sits among others like Rudin, Ahlfors, Sarason's notes, Herstein, though it's obvious to me that Artin is on a very very different path than all those books, very nonstandard (Artin DOES DO all that a usual algebra course does, and more, if you were wondering), but as a result, very very very thorough and very clearly presented. I love this book very much.

Rating:
Summary: Great book for challanging you to think with clarity
Review: Artin's book is probably one of the better books, more because of the way you have to read it to learn it. Artin's book is extremely nonstandard, in the sense that it isn't so "encyclopedic" as you usually encounter with the whole theorem, corollary, proof, proof, proof, example, example sequence. What I think a lot of readers miss is that Artin's book makes you fill in the details he leaves out by using the hints he mentions in words within the text. For example, I was able to expand the two pages of notes on Ch 2, section 5, in Artin into about 8 pages of original notes and theorems, just by digging for the main points. If you want a sample of my notes, please email me and I'll email you a brief PDF sample for you to compare. That being said, assume that you will have to dig a lot in this book, and should you choose to study from it, I suggest the following:

How to read it:

With a cup of coffee, or tea, and a notepad of paper for you to make comments on. Do not take notes; anyone knows that simply rewriting things doesn't do anything for learning. You should do the proofs in different ways, if you can see how, and try to make some of the aside remarks he makes into theorems or more precise ideas (this is not to say that Artin lacks rigor; this is just talking about the general commentary. When he makes commentary, it always seems to be enough to actually dig out exactly what to do after a little scratching). He also leaves a lot of easier proofs to the reader, so do them.

Is non-standard a less-rigorous approach?

No. Artin is definitely doing his own thing here, but I think it works really well. Getting through that book FORCES you to take responsibility for your math education by making you get your hands dirty while also developing an intuitive understanding of algebra.

What about his personal flavor of algebra?

Well, it's fairly clear to all of us that texts seem to have different flavors (being a function of the author's research area, and what was fashionable during the time the book was authored). Artin's book is algebra with light strong hints of geometry throughout, as he is in algebraic geometry. You will find that unlike most authors, Artin loves structures made of matrices when working with examples, as opposed to permutation groups or the ``symmetries of the square group,'' known also as the ``octic group.'' While these things have their place in his book, he changes the emphasis here. That's why I suggest using a companion book so as to have two sharply contrasting flavors of presentation, and Herstein seems to write in such a way that would do this. Artin covers a lot of material extremely quickly, but focuses on the bigger picture in several key areas. For example, the sections 7 and 8 in chapter 2 deal almost exclusively with how one would go about investigating a particular group structure to learn about it, teaching a student how to dig into something they might barely understand.

Advice to make a wondeful course:

Use another book which IS encyclopedic as a reference, since Artin doesn't label theorems and definitions so explicitly. I suggest Herstein's Abstract Algebra, or his book Topics in Algebra.

Personal Charracterization:

I place this book as one of my favorites on the bookshelf, and it sits among others like Rudin, Ahlfors, Sarason's notes, Herstein, though it's obvious to me that Artin is on a very very different path than all those books, very nonstandard (Artin DOES DO all that a usual algebra course does, and more, if you were wondering), but as a result, very very very thorough and very clearly presented. I love this book very much.

Rating:
Summary: Quite Simply the BEST
Review: By treating the concrete before the abstract, Artin has produced the clearest and easiest to understand expositon I have seen. He delves quite deeply into groups, rings, field theory and Galois theory. It is NOT true, as one reviewer claims, that Artin does not treat fields: an entire chapter is devoted to the topic.

If Bourbaki is your god and you believe axiomatization is the only way to present this material, then you won't like this book. But remember that this work is written by the son of the great Emil Artin, and Michael is a first-rate mathematician as well.

The ordering of topics and the approach are non-standard but this emphasis on the concrete before the abstract and the use of a function motivated development make this book stand apart from the competition. It is not only the best undergraduate abstract algebra text that I have seen but it can be very useful for graduate students. My undergraduate major was not in math, I HAD NO UNDERGRADUATE COURSE IN ABSTRACT ALGEBRA but I jumped into a really heavy-duty graduate level abstract algebra course with Hungerford as the text. Now, I feel that Dummit and Foote is much better than Hungerford and Artin is even better than the aforementioned and much better - and more thoughtful -than Gallian. I wish I had Artin to give me enlightenment and perspective when I was struggling with this material having had no prior exposure to it.

Rating:
Summary: good, solid treatment of algebra
Review: I bought this book for a class that I ended up dropping. In the beginning, I hated this book. I found Herstein's "topics in algebra" much better, and more to the point. It was only when I was getting bored with Herstein that I bothered to pick this up again. I was pleasantly surprised. A lot of the material flowed very smoothly - exactly as if Artin was teaching the material to you. It must however be noted that people tend to love or hate this book. This is predominantly due to the author's writing style. Given how expensive this book is, you might perhaps want to peruse it somewhere before deciding to buy it. But if you do, you'll get a solid exposition on most of the introductory topics in algebra as well as some insight on groups and symmetry, lie groups, representation theory, galois theory and quadratic number fields. And a whole lot of intuition as well, for the more regular topics. Give this book a chance - it's worth the effort and money.

Rating:
Summary: good, solid treatment of algebra
Review: I bought this book for a class that I ended up dropping. In the beginning, I hated this book. I found Herstein's "topics in algebra" much better, and more to the point. It was only when I was getting bored with Herstein that I bothered to pick this up again. I was pleasantly surprised. A lot of the material flowed very smoothly - exactly as if Artin was teaching the material to you. It must however be noted that people tend to love or hate this book. This is predominantly due to the author's writing style. Given how expensive this book is, you might perhaps want to peruse it somewhere before deciding to buy it. But if you do, you'll get a solid exposition on most of the introductory topics in algebra as well as some insight on groups and symmetry, lie groups, representation theory, galois theory and quadratic number fields. And a whole lot of intuition as well, for the more regular topics. Give this book a chance - it's worth the effort and money.

Rating:
Summary: Not a very good book
Review: I don't like this book because it doesn't follow a clear definition/theorem/proof format, like Rudin's Principles of Mathematical Analysis. In Artin's efforts to clarify he actually ends up confusing. The definitions and even some important concepts are haphazardly introduced in a "conversational" style, in the middle of long paragraphs (what are paragraphs doing in a math book?). These "explanatory" paragraphs are meant to clarify and add meaning, but although this style might sound helpful it actually gets in your way and makes learning much, much harder and more confusing. On the other hand, Rudin gets right to the point, he formalizes and crystallizes everything, he presents no pictures (and therefore forces you to make your own), he's not your friend, and in fact helps you learn better than Artin.

There are plentiful examples, but Artin's obsession with the rotation groups gets quite annoying; even if you're not interested in the rotation groups, you have to study them in-depth in the earlier chapters so that you can understand all the later examples, many of which refer back to these (e.g., the examples in the Group Representation chapter are almost entirely dependent on the tetrahedral group!). The examples in a high-level introductory book should be like those in Rudin's, throwing light on historically and theoretically important problems.

The exercises can range from hard and interesting to routine, with a pleasantly surprising number of the former. But Artin also seems to be obsessed with linear algebra and rotation groups; this can be frustrating. Also, some problems are just computationally hard--a consequence of the linear algebra fetish; there shouldn't be so many computational, almost-routine problems in a rigorous book.

(To those who don't know Rudin--I don't think he ever wrote an algebra text; I was only comparing presentation styles. Rudin does analysis.)

Rating:
Summary: A Bad Text Book
Review: I must read everything from any other text book to understand clearly what Artin is saying in his book. can you believe it? Few definition is described clearly, few theorem is proved in a logic-clear, easy-to-undersatnd way. The most important is that many useful properties of Ring , field are not included in his book, but in the problems you have to find all these totally by yourself in order to solve the problems. Also, the textbook is NOT well- orginized. A typical exmaple, Artin has not yet tell the reader basic informations and properties of Ring Of Polynomial in one variable , but he starts to describe the structure of Ring Of Polynomial in 2 or more variables. The reader's minds would be completely mixed up if he doesn't not have an extremely high IQ. I believe , your knowlege of Ring and Filed will be very limited and very unclear if you use Artin's book only. Abstract Algebra is a hard topic. You should not use a book which is definitely not a help for you but rather a trouble for you!.Artin may be a famous mathematician, but he is not a good educator. He doesn't not know to teach students in a good manner.

Rating:
Summary: It is a book
Review: I must read everything from Any other text book to understand clearly what Artin is saying in his book. can you believe it? Few definitions are described clearly, few theorems are proved in a logic-clear, easy-to-follow way. More importantly many useful properties of Ring, Field are not included in his book, but in the problems you have to find those out totally by yourself for the sake of solving the problems. Also, the textbook is NOT well- orginized. A typical exmaple, Artin has not yet tell the reader basic informations and properties of Ring of Polynomial in one variable, but he starts to describe the structure of Ring of Polynomial in 2 or more variables. The two are quite different. My mind was completely mixed up since I do not have extremely high IQ. I believe, your knowlege of Ring and Field will be very limited and remain unclear if you use Artin's book only. Abstract Algebra is a hard topic. You should not use a book which is definitely not a help for you but rather a trouble for you. Artin may be a famous mathematician, but he is not a good educator. He doesn't know how to coach students in a good manner.

Rating:
Summary: Exactly how an undergrad abstract algebra book should be
Review: Pretty much any introductory abstract algebra book on the market does a perfectly competent job of introducing the basic definitions and proving the basic theorems that any math student has to know. Artin's book is no exception, and I find his writing style to be very appropriate for this purpose. What sets this book apart is its treatment of topics beyond the basics--things like matrix groups and group representations. I suppose many introductory books shy away from much of the material on matrix groups in Artin's book because it involves a little analysis (and likewise for the section on Riemann surfaces in the chapter on field theory). However, Artin correctly realizes that a reasonably mathematically mature student--even one who doesn't know much analysis--will be able to profit from and enjoy the relatively informal treatments he gives these slightly more advanced topics. Of course these topics can also be found in graduate-level texts, but I for one would much rather be introduced to them via an example-based approach such as that in Artin than through the diagram-chasing obscurantism in more advanced books. I happened upon this book a little late--in fact, only after I'd taken a semester of graduate-level algebra and already felt like analysis was the path I wanted to take--but I'm beginning to think I would have been more keen on going into algebra if I'd first learned it from a book like this one.