This third edition of the standard text for modern algebra courses, teaches students as much about groups, rings and fields as possible in a first abstract algebra course with a minimum of introductory material on set theory. New chapters on isomorphism theorems, applications of G-sets to counting, free abelian groups, and a proof of the Jordan-Hoelder theorem have been added. Also included are sections of group action on a set, applications to burnside counting and the Sylow theorems with complete proofs.

A Very Few Preliminaries / Part I: Groups – Binary Operations / Groups / Subgroups / Permutations I / Permutations II / Cyclic Groups / Isomorphism / Direct Products / Finitely Generated Abelian Groups / Groups in Geometry and Analysis / Groups of Cosets / Normal Subgroups and Factor Groups / Homomorphisms / Series of Groups / Isomorphism Theorems; Proof of the Jordan-Hölder Theorem / Group Action on a Set /Applications of G-Sets of Counting / Sylow Theorems / Applications of the Sylow Theory / Free Abelian Groups / Free Groups / Group Presentations / Part II: Rings and Fields – Rings / Integral Domains / Some Noncommutative Examples / The Field of Quotients of an Integral Domain / Our Basic Goal / Quotient Rings and Ideals / Homomorphisms of Rings / Rings of Polynomials / Factorization of Polynomials over a Field / Unique Factorization Domains / Euclidean Domains / Gaussian Integers and Norms / Introduction to Extension Fields / Vector Spaces / Further Algebraic Structures / Algebraic Extensions / Geometric Constructions / Automorphisms of Fields / The Isomorphism Extension Theorem / Splitting Fields / Separable Extensions / Totally Inseparable Extensions / Finite Fields / Galois Theory / Illustrations of Galois Theory / Cyclotomic Extensions / Insolvability of the Quintic / Appendix / Bibliography / Answers and Comments / Notations / Index.

Undergraduate students