Waveland Press - Abstract Algebra, Fourth Edition, by John A. Beachy, William D. Blair

Highly regarded by instructors in past editions for its sequencing of topics and extensive set of exercises, the latest edition of Abstract Algebra retains its concrete approach with its gentle introduction to basic background material and its gradual increase in the level of sophistication as the student progresses through the book. Abstract concepts are introduced 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.

The fourth edition includes a new chapter of selected topics in group theory: nilpotent groups, semidirect products, the classification of groups of small order, and an application of groups to the geometry of the plane.

Students can download solutions to selected problems here.

Additional information, including a Study Guide for students, can be found on the authors' webpage.

"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

"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

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

10. Groups: Selected Topics
Nilpotent Groups / Internal Semidirect Products of Groups / External Semidirect Products of Groups / Classification of Groups of Small Order / The Orthogonal Group O₂(R) / Isometries of R²

Appendix
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