Instructors Preface
Students Preface
Dependence Chart
Sets and Relations
Ⅰ GROUPS AND SUBGROUPS
Introduction and Examples
Binary Operations
Isomorphic Binary Structures
Groups
Subgroups
Cyclic Groups
Generating Sets and Cayley Digraphs
Ⅱ PERMUTATIONS， COSETS， AND DIRECT PRODUCTS
Groups of Permutations
Orbits， Cycles， and the Alternating Groups
Cosets and the Theorem of Lagrange
Direct Products and Finitely Generated Abelian Groups
Plane Isometries
Ⅲ HOMOMORPHISMS AND FACTOR GROUPS
Homomorphisms
Factor Groups
Factor-Group Computations and Simple Groups
Group Action on a Set
Applications of G-Sets to Counting
Ⅳ RINGS AND FIELDS
Rings and Fields
Integral Domains
Fermats and Eulers Theorems
The Field of Quotients of an Integral Domain
Rings of Polynomials
Factorization of Polynomials over a Field
Noncommutative Examples
Ordered Rings and Fields
Ⅴ IDEALS AND FACTOR RINGS
Homomorphisms and Factor Rings
Prime and Maximal Ideals
Grobner Bases for Ideals
Ⅵ EXTENSION FIELDS
Introduction to Extension Fields
Vector Spaces
Algebraic Extensions
Geometric Constructions
Finite Fields
Ⅶ ADVANCED GROUP THEORY
Isomorphism Theorems
Series of Groups
Sylow Theorems
Applications of the Sylow Theory
Free Abelian Groups
Free Groups
Group Presentations
Ⅷ GROUPS IN TOPOLOGY
Simplicial Complexes and Homology Groups
Computations of Homology Groups
More Homology Computations and Applications
Homological Algebra
Ⅸ FACTORIZATION
Unique Factorization Domains
Euclidean Domains
Gaussian Integers and Multiplicative Norms
Ⅹ AUTOMORPHISMS AND GALOIS THEORY
Automorphisms of Fields
The Isomorphism Extension Theorem
Splitting Fields
Separable Extensions
Totally Inseparable Extensions
Galois Theory
Illustrations of Galois Theory
Cyclotomic Extensions
Insolvability of the Quintic
Appendix： Matrix Algebra
Bibliography
Notations
Answers to Odd-Numbered Exercises Not Asking for Definitions or Proofs
Index
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