This text introduces students to the algebraic concepts of group and rings, providing a comprehensive discussion of theory as well as a significant number of applications for each.
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 o 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.
New to This Edition
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.
Table of Contents Chapter 1: Number Theory Chapter 2: Groups I Chapter 3: Commutative Rings I Chapter 4: Linear Algebra Chapter 5: Fields Chapter 6: Groups II Chapter 7: Commutative Rings III