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Algebra: Chapter 0
Chapter 0 is a self-contained introduction to the main topics of algebra, suitable for a first sequence on the subject at the beginning graduate or upper undergraduate level. The primary distinguishing feature of the book, compared to standard textbooks in algebra, is the early introduction of categories, used as a unifying theme in the presentation of the main topics. A second feature consists of an emphasis on homological basic notions on complexes are presented as soon as modules have been introduced, and an extensive last chapter on homological algebra can form the basis for a follow-up introductory course on the subject. Approximately 1,000 exercises both provide adequate practice to consolidate the understanding of the main body of the text and offer the opportunity to explore many other topics, including applications to number theory and algebraic geometry. This will allow instructors to adapt the textbook to their specific choice of topics and provide the independent reader with a richer exposure to algebra. Many exercises include substantial hints, and navigation of the topics is facilitated by an extensive index and by hundreds of cross-references.
728 pages, Hardcover
First published August 2, 2009
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Displaying 1 - 6 of 6 reviews
March 13, 2019
This, to me, is the best book there is in general abstract algebra, perhaps even in any subject. Perhaps it is best left to a second course on the subject as even the author recommends, however, one requires next to no knowledge to follow. It is extremely clear, certainly aided by the natural concept of category theory, constantly emphasized along the whole journey. It is one of the few books that if I need a small refresher or catch-up on a subject or definition, I can go back to it and easily understand it better than in any other source at a fast pace. Whenever I'm reading something algebraic, I always make sure to have the book ready to be cracked open in the background.
There are tons of exercises, ranging from small and easy ones to practice the definitions, to hard ones such as proving Lagrange's theorem on sums of squares, however, they are often broken down into smaller ones with plenty of hints given, making it a very pleasant read and an extremely broad exposition. In fact, they follow a similar philosophy to Atiyah-Macdonald's Intro to Commutative Algebra, so that in only 700 or so pages, you can get the equivalent of a first and second course in abstract algebra, a course in commutative algebra, a Galois theory course, and a course on advanced linear algebra and homological algebra, with small applications to number theory, algebraic geometry, among others.
I genuinely don't think I will ever encounter another book like this, that you can read almost like a novel, and it's quite sad. I cannot recommend it enough to anyone that wishes to hear about it.
There are tons of exercises, ranging from small and easy ones to practice the definitions, to hard ones such as proving Lagrange's theorem on sums of squares, however, they are often broken down into smaller ones with plenty of hints given, making it a very pleasant read and an extremely broad exposition. In fact, they follow a similar philosophy to Atiyah-Macdonald's Intro to Commutative Algebra, so that in only 700 or so pages, you can get the equivalent of a first and second course in abstract algebra, a course in commutative algebra, a Galois theory course, and a course on advanced linear algebra and homological algebra, with small applications to number theory, algebraic geometry, among others.
I genuinely don't think I will ever encounter another book like this, that you can read almost like a novel, and it's quite sad. I cannot recommend it enough to anyone that wishes to hear about it.
July 30, 2020
fun + instructive + kind. I loved it. I still love it. I think I will love it forever.
February 16, 2022
This is the greatest textbook when it comes to learning undergraduate abstract algebra. Very well caliber for self learning and explained in both the formal and the non formal, giving the beginner a chance to grasp the concepts, often left for the reader to figure out or the teacher to present them in the context of a class in most mathematical expositions. I would direct computer scientist to this presentation of algebra and couldn't recommend it enough.
January 30, 2022
I didn't read the chapter on Fields or the Chapter on homological algebra. But based on the other chapters, I think its overall a very good text.
Pros:
Very clear explanations
Funny and enjoyable to read
Good exercises
Categories allow for stating many concepts more clearly and make the book seem more unified. Every time a new algebraic object is introduced, there will instantly be questions about the category they form appearing in your mind, which makes new objects seem less foreign, because these are questions you've answered for other categories.
The cons are basically the dual of the above. It is quite wordy. Also, the category stuff can be unwieldy if you don't have some mathematical maturity.
Pros:
Very clear explanations
Funny and enjoyable to read
Good exercises
Categories allow for stating many concepts more clearly and make the book seem more unified. Every time a new algebraic object is introduced, there will instantly be questions about the category they form appearing in your mind, which makes new objects seem less foreign, because these are questions you've answered for other categories.
The cons are basically the dual of the above. It is quite wordy. Also, the category stuff can be unwieldy if you don't have some mathematical maturity.
May 28, 2024
Why did bro use D_2n to represent the dihedral group 😫😫
April 8, 2023
Look no further if you have a genuine interest in mathematics in general and algebra in particular. It’s not light reading by any means, and I think it’s best appreaciated after a prior exposure to the subject in a more introductory manner, as the author himself suggests. But why?
First, the book is written with a reader that has acquaired some mathematical maturity. Even though it starts from the basics (as the title suggests, this is Chapter 0), but it cuts much deeper, at a faster pace, and puts more demands on the reader. For example, results that in intro texts are proved in the main body of the text here are left as exercises.
And second, I believe the categorical angle is best appreciated after a first brush through the subject. For this end, I cannot recommend enough Aluffi’s own Algebra Notes from the underground as a first contact with the subject. After that, you’ll have the background to undertake Chapter 0.
My only caveat (so far), is the fact that there’s not yet a follow up Chapter 1 in the horizon. I wish professor Aluffi comes to write it one day, as he suggests in the preface. I hope it won’t take him as long George RR Martin to write Winds of Winter though.
First, the book is written with a reader that has acquaired some mathematical maturity. Even though it starts from the basics (as the title suggests, this is Chapter 0), but it cuts much deeper, at a faster pace, and puts more demands on the reader. For example, results that in intro texts are proved in the main body of the text here are left as exercises.
And second, I believe the categorical angle is best appreciated after a first brush through the subject. For this end, I cannot recommend enough Aluffi’s own Algebra Notes from the underground as a first contact with the subject. After that, you’ll have the background to undertake Chapter 0.
My only caveat (so far), is the fact that there’s not yet a follow up Chapter 1 in the horizon. I wish professor Aluffi comes to write it one day, as he suggests in the preface. I hope it won’t take him as long George RR Martin to write Winds of Winter though.
Displaying 1 - 6 of 6 reviews