A Book of Abstract Algebra: Second Edition (Dover Books on Mathematics)

"A Book of Abstract Algebra" by Charles C. Pinter is widely regarded as the gold standard for self-learners, transforming a notoriously dense subject into an intuitive and deeply rewarding journey. Unlike traditional textbooks that present a dry "wall of theorems," Pinter adopts a conversational, almost Socratic approach where the exercises are carefully woven into the narrative, guiding you to discover the proofs and concepts yourself rather than just memorizing them. This "active learning" structure, combined with its incredibly affordable Dover price point, makes it a rare pedagogical masterpiece that builds genuine mathematical maturity and confidence without the intimidation factor of more rigid academic tomes. It's really a joy to learn from this book

As a mathematics student completing my second semester of an undergraduate abstract algebra sequence, my greatest lament is that the abstract algebra courses which I enrolled in used books which weren't completely satisfying. For group theory, we used Gallian's Contemporary Abstract Algebra , which isn't bad, but has significant room for improvement. For rings and fields, we used Herstein's Topics in Algebra , which probably wasn't bad when it was written 40 years ago, but is far too sparse on details, uses outdated or completely nonstandard notation, and contains six over-bloated (one entire chapter for ALL of group theory) and highly disorganized chapters. It wasn't until about midway through the semester on rings and fields that I discovered Pinter's text, and upon discovering it, I feel that I have found one of the nicest texts to use as a solid introduction to abstract algebra. Pinter doesn't skimp on any of the details, and not only fully explains everything, but is capable of explaining everything in a manner which is actually "easy" to understand, and dare I say, even enjoyable to read. But just because he explains it in a way that is "easy" to understand doesn't mean that the concepts themselves are easy. But with the way Pinter explains it, the concepts seem very natural. If I had to choose one thing that I like about Pinter's textbook, it would be the exercises which are provided at the end of each chapter. Just about every textbook out there has a highly disorganized hodgepodge of exercises, occasionally organized into categories like "concepts" and "theory" or marked with an asterisk for being more "difficult." Pinter takes an entirely different approach to the organization of exercises, by grouping related exercises under headings which summarize a group of them. This helps to develop even more theory than just what the book provides, without simply relegating important ideas to randomly numbered exercises which may be lost in a large set. It also helps students use a series of smaller proofs to establish important theorems rather than the approach that some others texts use, where they state a very important and sometimes "loaded" theorem and say that the "proof is left to the reader." There are also plenty of more basic conceptual exercises to build up to the theory. While some mathematical purists may snark at conceptual exercises, it is important to realize that understanding exactly what is going on at the most basic level, before delving into proofs, is very crucial. Certainly, theory and proofs are one of the most important things for a student to learn from an abstract algebra course, but how well can a student possibly prove properties of a coset if they can't even develop basic properties and examples of cosets? These exercises are also useful in preparation for exams, such as the GRE Mathematics Subject test... a multiple choice test that many undergrads going into grad school take which covers many topics, including abstract algebra. In a way, it is a shame that this has transitioned to a Dover text, as many professionals don't take Dover texts as seriously as texts from the "pushy" publishers cranking out the most up-to-date (or at least updating the cover, renumbering exercises to throw off students, and enhancing books with useless links to websites) books at 10 times the price. But this one is one that should be taken seriously, and even though it is just a reprint of a 20 year old text, it still has plenty of life in it for the time being. At the same time, though, it is nice that such a high quality text is being made available at such a low price, and even if professors aren't necessarily going to jump on the bandwagon to use this text in their courses, I think students of abstract algebra, particularly those who may be interested in eventually engaging in research in the area, will find this text to make a great supplement to some of the "problematic" texts out there. This text is also great for self-study as well.

I'm usually wary of reading reviews for math books since some of them say things like, "These book is fantastic, easy to read and great presentation! I just use it for casual self study. btw I have a PhD in algebraic quantum nano topology or whatever." My background, to give you context: I did my undergrad in computer science and have competed in programming contests, so I'm fairly well-versed in discrete math and decent reasoning skills. I took multivariable calculus, linear algebra, and some statistics and discrete math, and one light topology course that I struggled with. However, I never took any of the "math major" courses in college such as abstract algebra or real/complex analysis, though I've always wanted to. I'm reading this book now for self-study since I've always wanted to learn these other branches of math. When looking things up online, I occasionally come across terms like groups, rings, fields, homomorphisms, isomorphisms, etc. which I honestly didn't know what they meant, but these form the basic concepts of abstract algebra. In case you're wondering what abstract algebra is all about, I'll give a short summary of how he explains it: Abstract algebra is the study of algebraic structures. What's an algebraic structure? It's a set of elements along with some operation defined on those elements. It's a very general notion that encompasses arithmetic, polynomials, matrices, and more. It's the study of the general (or abstract) properties of certain types of sets. The first half of the book is devoted to studying groups, which are the simplest sets with some sort of structure to them, then builds on top of them to explore more advanced topics. The book even goes into number theory, like prime factorization and Diophantine equations, and uses abstract algebra to show why certain geometric constructs are impossible with only a ruler and compass. This book is most definitely doable for self-study! Here's what I found: Pros: -Very easy to follow along. He explains concepts very clearly and has the occasional diagram, and he very rarely makes claims without proving them, though sometimes he'll refer to a result from an exercise. -Each chapter is surprisingly short, so it gives quite a satisfying feeling to finish a chapter that's just a few pages long. This is excluding the exercises, FYI. The base content of each chapter is enough just for the fundamentals and not much more. Cons: -The bulk of each chapter is the exercises. There's actually a lot for each chapter, so I haven't been doing them all, just eyeballing the ones that look interesting and doing those. I highly recommend doing as many as possible, however; I'm just being lazy. -There aren't really any solutions for the exercises except for a handful. I've never struggled on an exercise long enough to have to look it up online. I don't know if this implies the problems are mostly easy. Seeing as how this is the first abstract algebra book I've read and I've never done a course in it, I can't compare it to anything else, but the presentation certainly feels very logical and natural.

The book is great. It reads like a journey. But the seller sent a bad copy. Last 10 pages had a 2 inch long cut. Since this was an international order, I didn't bother returning it, but I'm sorely disappointed in the seller A2 US.

Rich in concepts - directly introduced; easy to understand. An amazing book

I now love books published by Dover, they're cheap and brilliant. This book is no exception, it is short and light and will go up to (and slightly beyond) any second year undergraduate maths course. After that you're looking at the thicker "Springer Graduate Series" hardbacks really. I like this book because it has some great pictures, it's easy to just pick up and read and also the questions. Most of the book is questions. I wouldn't recommend this as your only abstract algebra book though. As always with books the first 1 or 2 chapters are mind-numbingly boring and tedious but after that it gets good. I would recommend another "hand-holding" book (write me a comment if you want to know what it is, I think it's called "essentials of abstract algebra" - ask and I will confirm) along side this, for if you are like me and at first lack confidence all those questions (DIY theorems if you will) can be daunting. However this book will both develop your confidence and provide you plenty of practice. I have absolutely no hesitation in recommending this book.

Excelente libro, tiene una estructura más similar a la de una conversación o una clase, lo que lo hace amigable, pero no se olvida de darnos demostraciones y ejercicios que dejan ver el poder del álgebra abstracta. Pero, para quienes estamos acostumbrados a los libros del tipo teorema - demostracion - ejemplo al principio puede ser extraño seguir la lectura de este libro

The important thing is that I think this book is the best in class for people who: - Perhaps at a first year university/ HS graduate level of mathematics. - Enjoys the motivation of concepts and ideas - Learns through solving problems If, however, you would like something that would: - Be a complete reference for abstract algebra - Have a lot of rigour in its exposition - Have challenging ideas that go very in-depth. then this is not the book for you. I wanted to write this review because it isn't really that popular as a candidate for abstract algebra, but I worked through this book at the start of this year and now after comparing to other abstract algebra books (D&F, Jacobson, etc) I find that this book occupies a niche that the other books don't capture. The writing of the book approaches algebra with the emphasis on motivation, but more importantly, beauty and natural ideas. It's really important for a person, which perhaps is not as mature in writing math to see why things work the way they are. One other resource that people often recommend to use as a gentle approach to algebra is 'Visual group theory', alongside the video series; however, I find that the ideas given are not as cohesive (and also i personally found the visualisations to be unhelpful when learning considering after a certain point it is important to understand to abstraction at hand). I especially want to highlight the route of "rings -> polynomial rings -> factorisation -> fields" that really motivates each concept through the previous. I see this as similar to the work of the more expository youtube videos (3b1b etc) where there is no real resource out there like that for abstract algebra. The exercises are plentiful and perhaps too computational in nature, but also for an inexperience learner computation does help solidify ideas. This is both a short coming and an advantage. By outright making core concepts exercise sets you end up reinforcing/ discovering ideas about algebra, however somethings were just monotonous. As I indicated above, if you want something to reference and you know everything/ enjoy a more efficient style, then I recommend Jacobson or Milne online notes for group theory; which compared to Pinter is more abstract in its proofs and serves as a more complete package.