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

One of the best books I've ever read. I was able to do 100% of the exercises up to chapter 20 before I ran out of steam. It was a lot of fun. The book is so well paced, it keeps you going. Some of the exercises are challenging - a couple took me 3 or 4 days each until the "aha" moment, when I found a two line solution. If only I'd followed all the rules and axioms I might have got it earlier. This teaches the value of discipline. This book has the feel of a text lovingly crafted over 20 years, tested against 100s of students, revised over and over again, and real labor of love. It shows. Ignore the people who said the problems are too hard. They are just right. As I said this is the best textbooks I've ever read, it's truly amazing the care that went into it. But more than that, it lets you make mistakes, even forces you to make mistakes so you can learn. It's so good. Those people who say the problems are too hard are not the people who will make the most of this book. It's hard enough to make you think, but having tested all the problems in the first 20 chapters I can say without a doubt they are doable by anyone - but you will have to do the work. Jigsaw puzzles are hard, and so is this - but it can be done. It's so satisfying when you finally get the answer to the hard ones. I wonder if the hard ones were officially hard or just hard for me. Once I founnd the answer I was thinking that wasn't so hard. That's what's good about this book, it really exercises the brain without being impossible. EVERYTHING is there to do all the excercises, you just have to look for the clues, and follow the axioms, use the rules, remember the algebraic properties, apply the techniques you have learned and DO IT PROPERLY. PLus, abstract algebra is quite simple - complex concepts but quite simple once you get the hang of it. This teaches you not to rely on intuition all the time. That works for integers at the supermarket, but for algebraic objects it's best to remember the rules and how to apply them at all stages. If you follow a disciplined approach then seemingly impossible answers become easy. This book is a fascinating journey into mathematics - it makes you feel the math, not the pain. It's a great book for hobbiests, beginners, etc. Not for professionals, although it might be fun for them to go through it. The book is for abstract concepts but it gives a feel for how math works, what it is, and whether you should continue in the subject. An amazing book. Some may disagree with my review, but this is an honest review - I truly believe that if you have any interest in learning rings, groups, fields, etc and are a beginner, then you will love this book - or at least find it very useful. This is the first math book I ever found to be completely fun and motivating.

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.

"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

This is quite a good book for a first introduction to abstract algebra. The only other algebra book I’ve read in any detail is Fraleigh’s, and Pinter’s is written at a slightly lower level, both in the style of presentation and the mathematical content. The first 30 or so chapters are great and what could have been a major problem with the book—the relatively limited material covered in the chapters themselves—is made up for in the exercise sets, which add greatly to the theoretical material covered in the book. Don’t worry if you don’t think you’re up to proving them yourself; they’re broken down into manageable chunks, sometimes even excessively so that some might call it hand holding. Just about anything covered in a typical first course is included one way or another in this book, and with help available anytime online now, relegating material to the exercises isn’t especially troublesome. There are the usual typos and very minor errors that make their way into just about every math book (one such minor error made the cover!) but nothing too serious. Until... I was prepared to give it five stars until I got to the later chapters on Galois theory, which seem rushed and a bit sloppy compared to the rest of the book. I did some detective work and checked this edition against the first and I know what happened. In the first edition there were some minor errors in some of the material, for example for one theorem to work he needs to assume that two field extensions are contained in a common extension but he never does, so the theorem he states implies that all splitting fields are equal instead of just isomorphic. Worse, he states a fundamentally incorrect theorem—there’s even an obvious error in the proof—and it looks like he implicitly relies on it a few chapters later. He tries to fix these in the second edition but patching up proofs and inserting some sentences here and there in the chapters instead of redoing them entirely is hard; the result is a hybrid of the old incorrect presentation and the corrected one (to be clear, it's now all correct, it just could be cleaned up a bit). This is confusing and results in one theorem that is still technically incorrect (but at least easily and obviously fixable) and a confusing proof about polynomials solvable by radicals having solvable Galois groups. I’ll provide the details and some more errata in a comment for anyone interested, which should be everyone planning to go through the entire book. So the first 30 chapters are fantastic and the final 3 are not fantastic, but still good. In fact, I'm slightly editing my previous review that probably made the final chapters sound worse than they are.

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.

I am technical, and I like math, but I am not a math whiz. I need books that go at a nice relaxed pace, and which give me plenty of opportunities to work interesting problems. This book fits those requirements perfectly. The explanations are simple and concrete. Pinter excels at breaking down complicated concepts into very small, practically painless steps. The exercises are a critical part of this scheme. A lot of important material is developed there, so I am trying to work them all. Very few answers are in the back of the book, but the problems are not very hard. In my case I am using this as a textbook in a course. I stay two chapters ahead of the professor, so I am sort of going through the book without a guide. The lack of a guide is no problem; in fact, the book serves as a guide to the professor, rather than the usual vice versa. This is not a perfect book. The index is pretty crummy, and the many sections of exercises are not labeled in a way that lets you see what subjects are being developed. The book doesn't work well if you are trying to look something up. Also, it omits some material covered by other introductory books (which I see as an advantage, but others might not). This book is really optimized for teaching, and it's very good at that. Combine this with the incredible price, and you have a fabulous deal.

Excellent starting point for group theory. The introduction defines the scope and purpose of the topic extremely well. Pedagogically superb. Proofs are not always rigorous which is a plus for this level. Proofs should teach and the author uses them efficiently. The quantity of exercises is large and there are many easy level exercises that allow you to test your comprehension. Exercises are use to extend the material as well which is useful for the more motivated readers. Ample examples are provided when new concepts are introduced. I have to say that I believe this book is one of the best in its topics and scope. As others have warned the Kindle version has formatting problems. I am able to navigate these pretty easily and have not discovered many true mistakes (on the part of the author that is). A real novice could easily get confused and may not be able to mentally correct some typos on the fly. It is recommended that you get the book in this case. Grab the free kindle sample and that will help you decide. Also navigation through the book is different than the typical and not as efficient but once I learned what was needed it is tolerable. My rating is of the book (high) and does not factor in the Kindle version problems. Have fun with a well written book.

I'm really enjoying studing for the first time Abstract Algebra with this book. I like how the chapters are quite small, to the point and, in the same time, rigorous and really easy to understand. The author explains really well all the concepts! Don't be fooled by the 4-6 pages chapters, the content is explained really well. But be warned: take you time and do the exercices in the end of the chapter, all of them. A review down below complains that he didn't like that a part of the content of the chapter is 'hidden' in the exercices, but I'm quite digging it! All the tools that you need are really well explained in the chapter, so when you do the exercices you feel like you're exploring the subject and making discoveries on your own, so you have this really nice feeling of satisfaction and accomplishment! And the way the exercices are layed down is great to: they are well leveled in terms of a difficult curve and there is plenty of help, insights and hints by the author alongside them. So you don't feel like the author just gave some exercices and a "good luck, see you in the next chapter", you feel his presence there and his orientation throughout them. I'm really impressed by the way this book was build and recommend to anyone who wants to learn Abstract Algebra for the first time. But remember, do the exercices, more than half the book is there (and like I explained, it's not a bad thing).

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.