Algebra

One of the chapters in Artin's book has a quote from Hermann Weyl: "In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics." If you've studied undergraduate algebra with any other book and then encountered this wonderful book, you'll understand what he meant. While Artin provides a comprehensive treatment of introductory algebra, starting with the most basic concepts, he covers a tremendous range of topics including matrix (Lie) groups and representation theory and Riemann surfaces. This does come at the expense of the usually comprehensive treatment of the standard topics of undergraduate algebra - readers hoping for ample opportunity to apply the Sylow classification theorems to describe all groups of order less than 100 or to describe the Galois groups of fourth-order polynomials will be sorely disappointed. Nonetheless, the reader mathematically mature enough to view these exercises as annoying as factoring polynomials was in high school algebra will appreciate this book. Artin's clear biases towards representation theory and algebraic geometry are obvious, but considering modern research in these fields is more active than in, say, the classification of finite groups or in Galois theory, this treatment makes sense. While some of the topics are more advanced than normally taught at the undergraduate level, the purpose of the book isn't to teach the method of mathematical proof but to provide a flavor of algebra and more importantly, its applications to other fields. Some of the problems are trivial (or at least easy), and emphasis is on intuition - Artin would rather you be able to visualize the action of a low-order dihedral group and its symmetries than to list and classify the subgroups of higher-order groups up to isomorphism. While this is an important exercise, it's the advanced-undergraduate equivalent of (high-school) polynomial factorization - it neither reinforces the heart of the subject nor does it inspire further interest in it. Besides, students contemplating graduate study in math or physics shouldn't be wasting valuable time practicing the Sylow theorems or working through proofs of the Isomorphism Theorems for every algebraic structure. Artin's book clearly represents a change in the way modern algebra should first be taught, and consequently makes the subject interesting - a difficult thing to do for what is typically a subject used only as a vehicle to teach basic techniques and mathematical rigor. For that reason it's not only the best book for the level, but one of the best math books ever written, and should be required not only by good math students but by physics students as well.

Okay so I guess this review is kind of biased - I've just completed the 2-semester algebra sequence at MIT, being taught by Prof. Artin himself. The course itself was truly fantastic, and throughout the year I have been in utter awe of Prof. Artin - he is without doubt as close to a role-model as I'm ever going to have. This is definitely a non-standard text; the approach is different to every single other algebra book I've read. It's tone is chatty, informal and eminently readable - a major bonus for me. I like to be able to read a text, say, before bed, and the Theorem-Proof-Corollary-Example-Theorem-Theorem style of many other books, coupled with the all to common 'line n+1 does not remotely follow from line n' syndrome that blights virtually every other book on the subject makes the rest (for me, anyway), unreadable without pen, paper and far more mental acuity than I usually have before bed. The choice of material is fantastic - whilst the coverage is not quite as encyclopedic as say, Dummit and Foote, the exposition is far clearer and far more relevant (at least as an undergraduate) to the average mathematician. That said, there is also a collection of genuinely fascinating topics you won't find in any other text at this level (the geometry of SU2, or function fields, for example) that add breadth and depth. A word on the exercises - excellent! Unlike Dummit and Foote (for instance) where the majority of the exercises are simply pen-pushing there are some extremely challenging yet extremely interesting exercies that go far beyond the average questions. That said, there are also a large number of interesting simple exercises - all of which are different (unlike various other books, which have say, 40 identical 'factor this polynomial...' or 'compute this galois group...' exercises in supreme boredom) and all of which are fun. I also like the way Prof. Artin never uses extremely annoying short cuts in proofs 'As you will show in the exercises...' or 'this step follows from exercise 42 on page 354' - whilst the exercises are not optional to anyone wanting to actually learn the material the style and flow of the exposition is not ruined by constant reference to them. The general style of the book is first to state, and intuitively describe the theorem in question, carefully making sure to motivate it well, then to give illuminating (and non-trivial) examples illustrating the power of the result in question, before finally giving a completely rigorous proof. Whilst the tone is chatty, and at a first glance the text may not seem entirely rigorous believe me the proofs themselves are perfect - short, succinct and most of all, beautiful. So yes, I genuinely believe this is the best algebra (and indeed, in any area of undergraduate mathematics) book I've ever read - I couldn't recommend it more highly.

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.

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:

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.