Abstract Algebra:  by John A. Beachy, William D. Blair
484 pages, $81.95 list
Instructor’s Solutions Manual available
Student and instructor resource material available here
Abstract Algebra
Third Edition
Highly regarded by instructors in past editions for its sequencing of topics as well as its concrete approach, slightly slower beginning pace, and extensive set of exercises, the latest edition of Abstract Algebra extends the thrust of the widely used earlier editions as it introduces modern abstract concepts only after a careful study of important examples. Beachy and Blair’s clear narrative presentation responds to the needs of inexperienced students who stumble over proof writing, who understand definitions and theorems but cannot do the problems, and who want more examples that tie into their previous experience. The authors introduce chapters by indicating why the material is important and, at the same time, relating the new material to things from the student’s background and linking the subject matter of the chapter to the broader picture.

Instructors will find the latest edition pitched at a suitable level of difficulty and will appreciate its gradual increase in the level of sophistication as the student progresses through the book. Rather than inserting superficial applications at the expense of important mathematical concepts, the Beachy and Blair solid, well-organized treatment motivates the subject with concrete problems from areas that students have previously encountered, namely, the integers and polynomials over the real numbers.

Supplementary material for instructors and students available on the book’s Web site: www.math.niu.edu/~beachy/abstract_algebra/
“I like the style of writing—formal and rigorous but not stuffy. It reads as an upper-level undergraduate text should. The exercises are the main reason I am interested in this book. They are a great mix of straightforward practice, some applications, and a healthy amount of theory that occasionally dives extra deep.” — Matt Keotz, Nazareth College

“The book has a great selection of exercises. There are enough good ones to make it possible to use the book several semesters in a row without repeating too much. They come in a nice mix from easy computations to warm the students up to more difficult theoretical problems. A few have solutions, but most don’t. I like this balance very much.” — Will Murray, California State University, Long Beach

“I like the gradual introduction to abstraction by starting with examples rather than abstract groups or rings. Many nice examples, as well as good theorems often omitted from undergraduate courses.” — William M. McGovern, University of Washington

“A well-written text with plenty of opportunity for students to get involved in the learning process.” — Francis T. Hannick, Minnesota State University, Mankato

“This book moves from concreteness to abstraction more skillfully than any text I have ever seen. A completely convincing and student-oriented presentation of the ‘why’ of abstract algebra as well as the ‘how.’” — Vic Camillo, University of Iowa
Table of Contents
1. Integers
Divisors / Primes / Congruences / Integers Modulo n

2. Functions
Functions / Equivalence Relations / Permutations

3. Groups
Definition of a Group / Subgroups / Constructing Examples / Isomorphisms / Cyclic Groups / Permutation Groups / Homomorphisms / Cosets, Normal Subgroups, and Factor Groups

4. Polynomials
Fields; Roots of Polynomials / Factors / Existence of Roots / Polynomials over Z, Q, R, and C

5. Commutative Rings
Commutative Rings; Integral Domains / Ring Homomorphisms / Ideals and Factor Rings / Quotient Fields

6. Fields
Algebraic Elements / Finite and Algebraic Extensions / Geometric Constructions / Splitting Fields / Finite Fields / Irreducible Polynomials over Finite Fields / Quadratic Reciprocity

7. Structure of Groups
Isomorphism Theorems; Automorphisms / Conjugacy / Groups Acting on Sets / The Sylow Theorems / Finite Abelian Groups / Solvable Groups / Simple Groups

8. Galois Theory
The Galois Group of a Polynomial / Multiplicity of Roots / The Fundamental Theorem of Galois Theory / Solvability by Radicals / Cyclotomic Polynomials / Computing Galois Groups

9. Unique Factorization
Principal Ideal Domains / Unique Factorization Domains / Some Diophantine Equations

Sets / Construction of the Number Systems / Basic Properties of the Integers / Induction / Complex Numbers / Solution of Cubic and Quartic Equations / Dimension of a Vector Space