抽象代数基本教程：第7版

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

本书是一部介绍抽象代数的入门书籍。假定学生了解了微积分和线性代数，并且理论大都在书中以例子和练习的形式出现。该书旨在教给学生尽可能多的群，环，以及域理论，最大特点是包含了较多的扎实的基础部分，这些部分对于更进一步的学习代数是有很大的帮助的。为了满足更多读者的要求，本书包含了很多有关拓扑中的同调群，同调群的计算以加深对因子群的理解。书中内容浅显易懂，为了将同调群讲述的更加清楚，在第6版的基础上减少了自动机，二进制线性密码以及部分代数结构。书后面附有不少练习，这些加深学生对内容的理解。
　　目次：群与子群；排列，陪集和直积；同态和因子群；环和域；理想和因子环；扩展域；高等群论；拓扑群；因子分解；自同态和伽罗瓦理论。