代数（英文版）|经典原版书库|（美）Michael Artin|机械工业出版社

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代数（英文版）

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原书名：Algebra	原出版社：Prentice Hall/Pearson
作者：（美）Michael Artin	出版社：机械工业出版社
译者：	丛书名：经典原版书库
出版日期：2004-3-1	上架日期：2005-10-8
ISBN：7111139135	页数：版次：1-3
开本：16开	装帧：
读者对象：本科生、研究生

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内容简介
本书由著名代数学家与代数几何学家Michael Artin所著，是作者在代数领域数十年的智慧和经验的结晶。书中既介绍了矩阵运算、群、向量空间、线性变换、对称等较为基本的内容，又介绍了环、模型、域，伽罗瓦理论等较为高深的内容，本书对于提高数学理解能力。增强对代数的兴趣是非常有益处的。此外，本书的可阅读性强，书中的习题也很有针对性，能让读者很快地掌握分析和思考的方法。
本书在麻省理工学院、普林斯顿大学、哥伦比亚大学等著名学府得到了广泛采用，是代数学的经典教材之一。

目录
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Preface
A Note for the Teacher
Chapter I  Matrix Operations
1.  The Basic Operations  1
2.  Row Reduction  9
3.  Determinants 18
4.  Permutation Matrices  24
5.  Cramer's Rule  28
EXERCISES 31
Chapter 2  Groups
1.  The Definition of a Group  38
2.  Subgroups  44
3.  Isomorphisms  48
4.  Homomorphisms  51
$.  Equivalence Relations and Partitions  53
6.  Cosets  57
7.  Restriction of a Homomorphism to a Subgroup  59
8.  Products of Groups 61
9.  Modular Arithmetic  64
10.  Quotient Groups 66
EXERCISES  69
Chapter 3  Vector Spaces
1.  Real Vector Spaces  78
2.  Abstract Fields  82
3.  Bases and Dimension  87
4.  Computation with Bases  94
5.  Infinite-Dimensional Spaces  100
6.  Direct Sums  102
EXERCISES  104
Chapter 4  Linear Transformations
1.  The Dimension Formula  109
2.  The Matrix of a Linear Transformation  111
3.  Linear Operators and Eigenvectors  115
4.  The Characteristic Polynomial  120
5.  Orthogonal Matrices and Rotations  123
6.  Diagonalization  130
7.  Systems of Differential Equations  133
8.  The Matrix Exponential  138
EXERCISES 145
Chapter 5  Symmetry
1.  Symmetry of Plane Figures  155
2.  The Group of Motions of the Plane  157
3.  Finite Groups of Motions  162
4.  Discrete Groups of Motions  166
5.  Abstract Symmetry: Group Operations  175
6.  The Operation on Cosets  178
7.  The Counting Formula  180
8.  Permutation Representations  182
9.  Finite Subgroups of the Rotation Group  184
EXERCISES  188
Chapter 6  More Group Theory
1.  The Operations of a Group on Itself  197
2.  The Class Equation of the Icosahedral Group  200
3.  Operations on Subsets  203
4.  The Sylow Theorems  205
5.  The Groups of Order 12  209
6.  Computation in the Symmetric Group  211
7.  The Free Group  217
8.  Generators and Relations  219
9.  The Todd-Coxeter Algorithm 223
EXERCISES  229
Chapter 7  Bilinear Forms
1.  Definition of Bilinear Form  237
2.  Symmetric Forms: Orthogonality  243
3.  The Geometry Associated to a Positive Form  247
4.  HermitianForms  249
5.  The Spectral Theorem 253
6.  Conics and Quadrics  255
7.  The Spectral Theorem for Normal Operators  259
8.  Skew-Symmetric Forms  260
9.  Summary of Results, in Matrix Notation 261
EXERCISES  262
Chapter 8  Linear Groups
1.  The Classical Linear Groups  270
2.  The Special Unitary Group SU2  272
3.  The Orthogonal Representation of SU2  276
4.  The Special Linear Group SL2(R)  281
5.  One-Parameter Subgroups  283
6.  The Lie Algebra 286
7.  Translation in a Group  292
8.  Simple Groups  295
EXERCISES  300
Chapter 9  Group Representations
1.  Definition of a Group Representation  307
2.  G-Invariant Forms and Unitary Representations  310
3.  Compact Groups  312
4.  G-Invariant Subspaces and Irreducible Representations
5.  Characters  316
6.  Permutation Representations and the Regular
Representation  321
7.  The Representations of the Icosahedral Group  323
8.  One-Dimensional Representations  325
9.  Schur's Lemma, and Proof of the Orthogonality
Relations 325
10.  Representations of the Group SU2  330
EXERCISES  335
Chapter 10 Rings                                      345
1.  Definition of a Ring  345
2.  Formal Construction of Integers and Polynomials  347
3.  Homomorphisms and Ideals  353
4.  Quotient Rings and Relations in a Ring  359
5.  Adjunction of Elements  364
6.  Integral Domains and Fraction Fields  368
7.  Maximal Ideals  370
8.  Algebraic Geometry  373
EXERCISES 379
Chapter 11 Factorization                                389
1.  Factorization of Integers and Polynomials  389
2.  Unique Factorization Domains, Principal Ideal Domains,
and Euclidean Domains  392
3.  Gauss's Lemma  398
4.  Explicit Factorization of Polynomials 402
5.  Primes in the Ring of Gauss Integers  406
6.  Algebraic Integers  409
7.  Factorization in Imaginary Quadratic Fields  414
8.  Ideal Factorization  419
9.  The Relation Between Prime Ideals of R and Prime
Integers  424
10.  Ideal Classes in Imaginary Quadratic Fields  425
11.  Real Quadratic Fields  433
12.  Some Diophantine Equations 437
EXERCISES 440
Chapter 12 Modules
1.  The Definition of a Module  450
2.  Matrices, Free Modules, and Bases  452
3.  The Principle of Permanence of Identities  456
4.  Diagonalization of Integer Matrices  457
5.  Generators and Relations for Modules  464
6.  The Structure Theorem for Abelian Groups 471
7.  Application to Linear Operators  476
8.  Free Modules over Polynomial Rings 482
EXERCISES 483
Chapter 13 Fields
1.  Examples of Fields  492
2.  Algebraic and Transcendental Elements 493
3.  The Degree of a Field Extension  496
4.  Constructions with Ruler and Compass 500
5.  Symbolic Adjunction of Roots  506
6.  Finite Fields 509
7.  Function Fields  515
8.  Transcendental Extensions  525
9.  Algebraically Closed Fields  527
EXERCISES  530
Chapter 14 Galois Theory
1.  The Main Theorem of Galois Theory  537
2.  Cubic Equations  543
3.  Symmetric Functions  547
4.  Primitive Elements  552
5.  Proof of the Main Theorem  556
6.  QuarticEquations  560
7.  Kummer Extensions  565
8.  Cyclotomic Extensions  567
9.  QuinticEquations  570
EXERCISES 575
Appendix  Background Material
1.  Set Theory 585
2.  Techniques of Proof  589
3.  Topology  593
4.  The Implicit Function Theorem  597
EXERCISES  599
Notation
Suggestions for Further Reading
Index

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书中既介绍了矩阵运算、群、向量空间、线性变换、对称等较为基本的内容，又介绍了环、模型、域、伽罗瓦理论等较为高深的内容，本书对于提高数学理解能力、增强对代数的兴趣是非常有益处的。此外，本书的可阅读性强，书中的习题也很有针对性，能让读者很快地掌握分析和思考的方法。

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