Chapter 1: Number Theory Induction Binomial Coefficients Greatest Common Divisors The Fundamental Theorem of Arithmetic Congruences Dates and Days   Chapter 2: Groups I Some Set Theory Permutations Groups Subgroups and Lagrange's Theorem Homomorphisms Quotient Groups Group Actions Counting with Groups   Chapter 3: Commutative Rings I First Properties Fields Polynomials Homomorphisms Greatest Common Divisors Unique Factorization Irreducibility Quotient Rings and Finite Fields Officers, Magic, Fertilizer, and Horizons   Chapter 4: Linear Algebra Vector Spaces Euclidean Constructions Linear Transformations Determinants Codes Canonical Forms   Chapter 5: Fields Classical Formulas Insolvability of the General Quintic Epilog   Chapter 6: Groups II Finite Abelian Groups The Sylow Theorems Ornamental Symmetry   Chapter 7: Commutative Rings III 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.