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

by Charles C Pinter (Author)
4.5 4.5 out of 5 stars 533 ratings
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Accessible but rigorous, this outstanding text encompasses all of the topics covered by a typical course in elementary abstract algebra. Its easy-to-read treatment offers an intuitive approach, featuring informal discussions followed by thematically arranged exercises. Intended for undergraduate courses in abstract algebra, it is suitable for junior- and senior-level math majors and future math teachers. This second edition features additional exercises to improve student familiarity with applications.
An introductory chapter traces concepts of abstract algebra from their historical roots. Succeeding chapters avoid the conventional format of definition-theorem-proof-corollary-example; instead, they take the form of a discussion with students, focusing on explanations and offering motivation. Each chapter rests upon a central theme, usually a specific application or use. The author provides elementary background as needed and discusses standard topics in their usual order. He introduces many advanced and peripheral subjects in the plentiful exercises, which are accompanied by ample instruction and commentary and offer a wide range of experiences to students at different levels of ability.
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  1. ISBN-10
    0486474178
  2. ISBN-13
    978-0486474175
  3. Edition
    Second
  4. Publisher
    Dover Publications
  5. Publication date
    January 14, 2010
  6. Part of series
    Dover Books on Mathematics
  7. Language
    English
  8. Dimensions
    5.4 x 0.8 x 8.4 inches
  9. Print length
    400 pages
  10. See all details
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From the Publisher

algebra college

Accessible but rigorous

Outstanding text encompasses all of the topics covered by a typical course in elementary abstract algebra

Its easy-to-read treatment offers an intuitive approach, featuring informal discussions followed by thematically arranged exercises. Intended for undergraduate courses in abstract algebra, it is suitable for junior- and senior-level math majors and future math teachers. This second edition features additional exercises to improve student familiarity with applications.

Editorial Reviews

About the Author

Charles C. Pinter is Professor Emeritus of Mathematics at Bucknell University.

Product details

  • Publisher ‏ : ‎ Dover Publications; Second edition (January 14, 2010)
  • Language ‏ : ‎ English
  • Paperback ‏ : ‎ 400 pages
  • ISBN-10 ‏ : ‎ 0486474178
  • ISBN-13 ‏ : ‎ 978-0486474175
  • Item Weight ‏ : ‎ 2.31 pounds
  • Dimensions ‏ : ‎ 5.4 x 0.8 x 8.4 inches
  • Customer Reviews:
    4.5 4.5 out of 5 stars 533 ratings

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Charles C. Pinter

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Top reviews from the United States

Troy
5.0 out of 5 stars Amazing exposition
Reviewed in the United States on March 3, 2024
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Used this for self-study. The exposition in this book is excellent and extremely intuitive. The author does a great job explaining concepts rigorously and intuitively at the same time. Diagrams are excellent.

This book opened my mind to entirely new ways of thinking and I had fun reading it and making connections between the chapters. The book is organized very cleanly and is a good size. Highly recommend for a self-studier.
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Steven Skees
5.0 out of 5 stars Excellent book for the curious student
Reviewed in the United States on April 11, 2012
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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.
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Noonoh
5.0 out of 5 stars Great for self-study!
Reviewed in the United States on October 17, 2018
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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.
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Top reviews from other countries

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Soy yo
4.0 out of 5 stars Muy bueno
Reviewed in Mexico on July 29, 2022
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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
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N_P
5.0 out of 5 stars Aprendi Álgebra com esse Livro
Reviewed in Brazil on January 13, 2023
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Nem muito fácil, nem muito difícil. Cumpre o que promete. Aprendi algumas coisas novas. Dá pra aprender todo conteúdo de 3 cursos de álgebra: Grupos, Anéis e Teoria de Galois.

Porém acho que faltou alguns detalhes. Por exemplo, na parte de grupos ele toca apenas brevemente no Teorema de Sylow. E apenas como exercício opcional.
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Fabrizio
5.0 out of 5 stars Ottimo libro
Reviewed in Italy on January 16, 2023
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Ottimo libro
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J.B
5.0 out of 5 stars Reads like a novel
Reviewed in Germany on September 21, 2022
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Fantastic first book for abstract algebra!
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Avi C
5.0 out of 5 stars A great book for non-mathmaticians to learn beautiful pure math
Reviewed in Canada on September 6, 2019
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I found this book extremely helpful. The author does not presuppose any previous knowledge and does not over use mathematical notation. Many of the proofs and theorems are simply written out in crystal clear English prose with a few symbols that are defined along the way. He also includes austere but very helpful drawings to illustrate concepts (e.g., functions having the properties of injective, surjective, or bijective is shown with helpful intuitive dot and arrow diagrams).

I picked up this book because I developed a fascination with group theory after reading some lay-person books on the subject. I decided it was time to try to learn the real math because I was frustrated by the vagueness of the popular math books. It has been very difficult to find books on topics in pure math that don't presuppose a lot of previous university level math courses. I never took math beyond first year university level linear algebra, although I did study formal logic, and I work as a web developer so I have some facility for abstract reasoning.

My focus in reading this book has been to try to understand the ideas, I ignored most of the exercises and applications so I can’t speak to the quality of those. I think this book is excellent for any non-mathematician who wants to learn some more advanced mathematical ideas for the sake of their intrinsic beauty.
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