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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