A First Course in Abstract Algebra: Rings, Groups, and Fields, Third Edition, 3rd Edition (Hardback) book cover

A First Course in Abstract Algebra

Rings, Groups, and Fields, Third Edition, 3rd Edition

By Marlow Anderson, Todd Feil

Chapman and Hall/CRC

552 pages | 40 B/W Illus.

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Like its popular predecessors, A First Course in Abstract Algebra: Rings, Groups, and Fields, Third Edition develops ring theory first by drawing on students’ familiarity with integers and polynomials. This unique approach motivates students in the study of abstract algebra and helps them understand the power of abstraction. The authors introduce groups later on using examples of symmetries of figures in the plane and space as well as permutations.

New to the Third Edition

  • Makes it easier to teach unique factorization as an optional topic
  • Reorganizes the core material on rings, integral domains, and fields
  • Includes a more detailed treatment of permutations
  • Introduces more topics in group theory, including new chapters on Sylow theorems
  • Provides many new exercises on Galois theory

The text includes straightforward exercises within each chapter for students to quickly verify facts, warm-up exercises following the chapter that test fundamental comprehension, and regular exercises concluding the chapter that consist of computational and supply-the-proof problems. Historical remarks discuss the history of algebra to underscore certain pedagogical points. Each section also provides a synopsis that presents important definitions and theorems, allowing students to verify the major topics from the section.


"I am a fan of the rings-first approach to algebra, agreeing with the authors that students’ familiarity with the integers and with polynomials renders rings more intuitive and accessible than groups. But this book has many other virtues besides presenting the material in this order. For example, each section is preceded and followed by short sections that try to put the material into a broader context. … This is definitely a book worth considering for textbook adoption."

MAA Reviews, November 2014

Praise for the Second Edition:

"I was quickly won over by the book … . The book is very complete, containing more than enough material for a two semester course in undergraduate abstract algebra … . Even though there was a great deal of material presented, I found the book to be very well organized. … There are a lot of things that I like about this book. … [It is] well written and will help students to see the big picture. … All in all it seems that a lot of thought went into this book, resulting in a comprehensive, well-written, readable book for undergraduates first learning abstract algebra."

—MAA Online

"A remarkable feature of the book is that it starts first with the concept of a ring, while groups are introduced later. The reason of that is that students are usually more familiar with various number domains rather than the mappings and matrices. There is a huge number of examples in the book … . The book contains a lot of nice exercises of various degrees of difficulty so that it can also be used as a practice book."

EMS Newsletter, March 2006

Table of Contents

Numbers, Polynomials, and Factoring

The Natural Numbers

The Integers

Modular Arithmetic

Polynomials with Rational Coefficients

Factorization of Polynomials

Section I in a Nutshell

Rings, Domains, and Fields


Subrings and Unity

Integral Domains and Fields


Polynomials over a Field

Section II in a Nutshell

Ring Homomorphisms and Ideals

Ring Homomorphisms

The Kernel

Rings of Cosets

The Isomorphism Theorem for Rings

Maximal and Prime Ideals

The Chinese Remainder Theorem

Section III in a Nutshell


Symmetries of Geometric Figures


Abstract Groups


Cyclic Groups

Section IV in a Nutshell

Group Homomorphisms

Group Homomorphisms

Structure and Representation

Cosets and Lagrange's Theorem

Groups of Cosets

The Isomorphism Theorem for Groups

Section V in a Nutshell

Topics from Group Theory

The Alternating Groups

Sylow Theory: The Preliminaries

Sylow Theory: The Theorems

Solvable Groups

Section VI in a Nutshell

Unique Factorization

Quadratic Extensions of the Integers


Unique Factorization

Polynomials with Integer Coefficients

Euclidean Domains

Section VII in a Nutshell

Constructibility Problems

Constructions with Compass and Straightedge

Constructibility and Quadratic Field Extensions

The Impossibility of Certain Constructions

Section VIII in a Nutshell

Vector Spaces and Field Extensions

Vector Spaces I

Vector Spaces II

Field Extensions and Kronecker's Theorem

Algebraic Field Extensions

Finite Extensions and Constructibility Revisited

Section IX in a Nutshell

Galois Theory

The Splitting Field

Finite Fields

Galois Groups

The Fundamental Theorem of Galois Theory

Solving Polynomials by Radicals

Section X in a Nutshell

Hints and Solutions

Guide to Notation


Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Algebra / General