3 Review(s)

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. Table of Contents 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.