Algebra: Chapter 0 (Graduate Studies in Mathematics, 104)

In my experience, Algebra is the discipline that has the least variety in its textbook choices in all the major fields of upper-level undergraduate to graduate mathematics. For a long (but, hopefully, illustrative) for instance, a semester of analysis of a single real variable through the Riemann integral delivered through a healthy course of Rudin or Abbott can be followed by its legacy heirs in "advanced calculus" with Callahan, Pugh (which can be used for an excellent first half, as well!), the long-forgotten Bressoud "Second Year Calculus" text for those wishing to mix ad-cal material more with the sciences or for crowds with more the needs of a Lay-style text for the first semester, Munkres for the tougher crowd (whether topology or analysis on manifolds - why not both?), Ghorpade for an excellent application-focused follow-up, or even Johnston (or Rudin Sr. for graduates) for those wanting to go straight for the Kool-Aid and learn Lebesgue integration. Nearly any course works for a fun capstone that prepares you for your career (or career in grad school, as the case may be). Heck, even Complex Analysis and any one of the droves of courses whose book intros claim "all you need to know is the calculus and linear algebra you learned in second grade to begin!" This is not so with Algebra, and I wish I knew of what it was like before Lang's (oft unfairly) maligned pages shook the shelves of Algebra books (both literally and figuratively) and defined all that went after, with Herstein - I think - leading the way with undergraduate algebra (Herstein is opposite of Hungerford: his gradute volumes were better than his undergraduate ones). Pick up a Birkhoff and Mac Lane and compare it to Lang's capable yet dictionarian text, or Dummit and Foote with all its dryness, and then take it with you for something better to read as you go through your grad course - Jacobson's two volumes, both on Dover, would aslo work as companions. Pinter, which everyone should own and which tops teaching the subject over any ither volume (modulo course level), doesn't even stand apart as offering a different take. We have had nothing. Until Aluffi. Allufi has done to Algebra what Linear Algebra Done Right did to Linear Algebra: it took an important idea that shows up in typical texts in some extreme way and then recasts the idea in some equally extreme way, changing the tone of the whole subject in so doing. But, unlike Axler's great text, Aluffi's Algebra: Chapter 0 has the honor of a true unique take. Curing Dummit and Footte's appendicitis by taking it to the very front of the text (Chapter 0, he very well could have called it), the entire tone and pace of the book grows as you watch category theory's tools develop it to a status of near-omnipresence. The style is direct, but not Hungerford (graduate Hungerford) direct: it explains what it needs to explain, does it well, and does something almost no algebra book does: this book often gives straightforward reasons why results are important, both to areas from elsewhere in the text to completely outside of abstract algebra, and even gives hints - correct, as it turned out for me! - as to what appears on qualifying exams (e.g. the discussion preceding Claim 2.16 on page 201). The Linear Algebra chapters are fantastic - they aren't just "pretend it says vector spaces where it says free modules and cross your fingers" kinds of sections that follow the inevitable letter-pushing introduction. Category Theory, which has been growing with you the whole way, jumps in here in an important way, and though this is exactly where I currently am in my perusal of the book, I am quietly confident that the discussion on multilinear algebra and tensor products will go far better than it did under D&F. Nonetheless, I still have my copy of Grinfield and Bachman on hand to ground the discussion into reality (or at least some sense of it). The book then concludes with Homological Algebra, and given the author's style I'd be surprised if the old joke with Lang's Algebra is missing, the one where Lang's corresponding part has -as a real question - instructions for the reader to rent a book on homological algebra and then to prove every theorem and sove every exercise in the book. Since category theory has had such a role in this book, I have no doubt that the topic will find its resting place (in the context of this book, at least) in the grounds of homological algebra, and I'm excited to read this topic for the first time once I get there shortly. The book is a delight, and to my original question as to whether it should be "the new D&F" (Dummitt and Foote) only time will tell - having already taken algebra, my view is skewed, but imagining myself as a younger man facing graduate algebra for the first time, my answer tends toward "I hope so."

I should first mention that I, along with about twenty of my fellow first-year mathematics graduate students, scoured this book from beginning to end. We completed nearly every exercise, and discovered a number of errata (there is quite a large list available on the author's website, but this book shines in spite of it all). I've experienced Fraleigh, Artin, Dummit and Foote, and Aluffi's texts on abstract algebra. While each has it's place, I have to say that Aluffi is my favorite. His writing style is phenomenal (and humorously pretentious at times). This text is not intended to be a reference, but instead read from start to finish, and Aluffi monopolizes this to its full effect. The content is spot on for the intended audience. His exercises cover important, relevant topics to important fields I and my fellow graduate students intend to pursue. These include, but are not limited to: algebraic geometry, commutative algebra, homological algebra, and Lie theory. This book is the best I have encountered for transitioning from an elementary understanding of abstract algebra to a mature perspective, backed by the might of category theory. That being said, I can see how the book may go more smoothly if one has had some initial exposure to algebra before Aluffi. This text does an excellent job synthesizing my understanding, but the organization could be confusing for a beginner. My only real disappointment with the book is in the final chapter on homological algebra. By the last two or three sections, the content is almost prohibitively confusing. It could be the case that there are errata that have confused me (indeed, the listed errata on his website sharply fall in this chapter, and I believe it's because most students don't get this far). But more likely this last chapter could use some reorganization and more basic exercises early on. Of course, this is the newest and most mature topic covered in the book, it was my first time encountering the material, and it is probably intended to be a test of the reader's maturity developed throughout the book. In any event, it didn't click for me as well as the rest of the book. All in all, I'll definitely be purchasing the second edition, and I'm eagerly awaiting Chapter 1 :)

This book approaches a classical topic in a fresh way that allows synthesis of algebraic theory using and simultaneously motivating modern (categorical) terminology. The latter is often presented indirectly by jumping right in and concatenation logical constructs in vacuo. By attaching it to the algebra exposition the book is more pleasing and both topics are mutually enhanced.

Great book for learning category theory

This is a terrific book. I bought this book to learn about category theory, primarily to get a better handle on what's going on in Haskell. I'm not a professional programmer or a mathematician, just a dilettante, although I was a math major during my first attempt to get my degree. I did take some proof centered introductory courses in subjects like set theory, linear algebra, topology, and so on. I feel like people have different learning styles. Some people like to see examples and do problems, and build up a big picture understanding from the bottom up, while others like to start with the big picture, and have an idea of what the point of it all is from the beginning. I think that Lawvere and Schanuel's book is great for people who learn best with examples and and problems, while this book is about as good as it gets for people like me, who want to start with a bigger picture and then flesh it out. I don't know if anyone remembers my old texts, but this book seems to be similar in spirit to ones I liked the best. Saracino's Abstract Algebra: A First Course, Hoffman and Kunze's Linear Algebra, Kaplansky's Set Theory and Metric Spaces, Gamelin's Introduction to Topology, etc. I suspect that if you like those, you'll like Algebra: Chapter 0 a lot.

great book

Starts almost at the very beginning, and ends ... Well, I'll just say that I'm probably not going to manage the sequel, "Algebra: Chapter 1". A delight to read, reread, check, think about.