You are here

A First Course in Abstract Algebra: Rings, Groups, and Fields

Marlow Anderson and Todd Feil
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
, on

See our review of the second edition.

The most significant change made for the third edition is a reorganization of the material on unique factorization in domains, “reflecting that for many teaching a first course this topic is optional.” That required also reorganizing the material on ideals.

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. As the authors say, students first learning algebra tend to get lost in the details, so this attempt to make them notice the landscape rather than just focusing on the stones in their shoes is to be welcomed.

This is definitely a book worth considering for textbook adoption.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME, and the author of A Guide to Groups, Rings, and Fields.

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