ABSTRACT

Emphasizing active learning, this text not only teaches abstract algebra but also provides a deeper understanding of what mathematics is, how it is done, and how mathematicians think. The book can be used in both rings-first and groups-first abstract algebra courses. Numerous activities, examples, and exercises illustrate the definitions, theorems, and concepts. Each chapter also discusses the connections among topics in ring theory and group theory, helping students see the relationships between the two main types of algebraic objects studied throughout the text.

part I|2 pages

The Integers

chapter 1|8 pages

The Integers: An Introduction

chapter 2|12 pages

Divisibility of Integers

chapter 3|10 pages

Greatest Common Divisors

chapter 4|10 pages

Prime Factorization

part II|2 pages

Other Number Systems

chapter 5|18 pages

Equivalence Relations and Zn

chapter 6|14 pages

Algebra in Other Number Systems

part III|2 pages

Rings

chapter 7|12 pages

An Introduction to Rings

chapter 8|14 pages

Integer Multiples and Exponents

chapter 9|16 pages

Subrings, Extensions, and Direct Sums

chapter 10|14 pages

Isomorphism and Invariants

part IV|2 pages

Polynomial Rings

chapter 11|16 pages

Polynomial Rings

chapter 12|14 pages

Divisibility in Polynomial Rings

chapter 13|12 pages

Roots, Factors, and Irreducible Polynomials

chapter 14|20 pages

Irreducible Polynomials

chapter 15|16 pages

Quotients of Polynomial Rings

part V|2 pages

More Ring Theory

chapter 16|22 pages

Ideals and Homomorphisms

chapter 18|20 pages

From Z to C

part VI|2 pages

Groups

chapter 19|12 pages

Symmetry

chapter 20|12 pages

An Introduction to Groups

chapter 21|8 pages

Integer Powers of Elements in a Group

chapter 22|14 pages

Subgroups

chapter 23|8 pages

Subgroups of Cyclic Groups

chapter 24|8 pages

The Dihedral Groups

chapter 25|14 pages

The Symmetric Groups

chapter 26|12 pages

Cosets and Lagrange’s Theorem

chapter 27|22 pages

Normal Subgroups and Quotient Groups

chapter 28|12 pages

Products of Groups

chapter 29|26 pages

Group Isomorphisms and Invariants

chapter 30|14 pages

Homomorphisms and Isomorphism Theorems

chapter 32|14 pages

The First Sylow Theorem

chapter 33|10 pages

The Second and Third Sylow Theorems

part VII|2 pages

Special Topics

chapter 34|10 pages

RSA Encryption

chapter 35|10 pages

Check Digits

chapter 36|12 pages

Games: NIM and the 15 Puzzle