Induction

Binomial Coefficients

Greatest Common Divisors

The Fundamental Theorem of Arithmetic

Congruences

Dates and Days

Some Set Theory

Permutations

Groups

Subgroups and Lagrange's Theorem

Homomorphisms

Quotient Groups

Group Actions

Counting with Groups

First Properties

Fields

Polynomials

Homomorphisms

Greatest Common Divisors

Unique Factorization

Irreducibility

Quotient Rings and Finite Fields

Officers, Magic, Fertilizer, and Horizons

Vector Spaces

Euclidean Constructions

Linear Transformations

Determinants

Codes

Canonical Forms

Classical Formulas

Insolvability of the General Quintic

Epilog

Finite Abelian Groups

The Sylow Theorems

Ornamental Symmetry

Prime Ideals and Maximal Ideals

Unique Factorization

Noetherian Rings

Varieties

Grobner Bases

Hints for Selected Exercises

Bibliography

Index

• Comprehensive coverage of abstract algebra – Includes discussions of the fundamental theorem of Galois theory; Jordan-Holder theorem; unitriangular groups; solvable groups; construction of free groups; von Dyck's theorem, and presentations of groups by generators and relations.

• Significant applications for both group and commutative ring theories, especially with Gr ö bner bases –  Helps students see the immediate value of abstract algebra.

• Flexible presentation – May be used to present both ring and group theory in one semester, or for two-semester course in abstract algebra.

• Number theory – Presents concepts such as induction, factorization into primes, binomial coefficients and DeMoivre's Theorem, so students can learn to write proofs in a familiar context.

• Section on Euclidean rings – Demonstrates that the quotient and remainder from the division algorithm in the Gaussian integers may not be unique. Also, Fermat's Two-Squares theorem is proved.

• Sylow theorems – Discusses the existence of Sylow subgroups as well as conjugacy and the congruence condition on their number.

• Fundamental theorem of finite abelian groups – Covers the basis theorem as well as the uniqueness to isomorphism

• Extensive references and consistent numbering system for lemmas, theorems, propositions, corollaries, and examples – Clearly organized notations, hints, and appendices simplify student reference.

• Rewritten for smoother exposition – Makes challenging material more accessible to students.

• Updated exercises – Features challenging new problems, with redesigned page and back references for easier access.

• Extensively revised Ch. 2 (groups) and Ch. 3 (commutative rings ) – Makes chapters independent of one another, giving instructors increased flexibility in course design.

• New coverage of codes –  Includes 28-page introduction to codes, including a proof that Reed-Solomon codes can be decoded.

• New section on canonical forms (Rational, Jordan, Smith) for matrices – Focuses on the definition and basic properties of exponentiation of complex matrices, and why such forms are valuable.

• New classification of frieze groups – Discusses why viewing the plane as complex numbers allows one to describe all isometries with very simple formulas.

• Expanded discussion of orthogonal Latin squares – Includes coverage of magic squares.

• Special Notation section – References common symbols and the page on which they are introduced.