著名代数学家与代数几何学家Michael Artin所著的《代数(英文版)(第2版)》是一本代数学的经典著作，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、域、伽罗瓦理论等较为高深的内容，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。《代数》是一本有深度、有特点的著作，适合数学工作者以及基础数学、应用数学等专业的学生阅读。

《代数(英文版)(第2版)》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、线性算子、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。《代数(英文版)(第2版)》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，《代数(英文版)(第2版)》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。

作者结合这20年来的教学经历及读者的反馈，对本版进行了全面更新，更强调对称性、线性群、二次数域和格等具体主题。本版的具体更新情况如下：

新增球面、乘积环和因式分解的计算方法等内容，并补充给出一些结论的证明，如交错群是简单的、柯西定理、分裂定理等。

修订了对对应定理、SU2表示、正交关系等内容的讨论，并把线性变换和因子分解都拆分为两章来介绍。

新增大量习题，并用星号标注出具有挑战性的习题。

《代数(英文版)(第2版)》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。

Freface

1 Matrices

 1.1 The Basic Operations

 1.2 Row Reduction

 1.3 The Matrix Tianspose

 1.4 Determinants

 1.5 Permutations

 1.6 Other Formulas for the Determinant

 Exercises

2 Groups

 2.1 Laws of Composition

 2.2 Groups and Subgroups

 2.3 Subgroups of the Additive Group of Integers

 2.4 Cyclic Groups

 2.5 Homomorphisms

 2.6 Isomorphisms

 2.7 Equivalence Relations and Partitions

 2.8 Ccsets

 2.9 Modular Arithmetic

 2.10 The Correspondence Theorem

 2.11 Product Groups

 2.12 Quotient GrouFs

 Exercises

3 Vector Spaces

 3.1 Subspaces of Rn

 3.2 Fields

 3.3 Vector Spaces

 3.4 Bases and Dimension

 3.5 Computing with Bases

 3.6 Direct Sums

 3.7 Infinite-Dimensional Spaces

 Exercises

4 Linear Operators

 4.1 The Dimension Formula

 4.2 The Matrix of a Linear Transformation

 4.3 Linear Operators

 4.4 Eigenvectors

 4.5 The Characteristic Polynomial

 4.6 Triangular and Diagonal Fcrms

 4.7 Jordan Form

 Exercises

5 Applications of Linear Operators

 5.1 Orthogonal Matrices and Rotations

 5.2 Using Continuity

 5.3 Systems of Differential Equations

 5.4 The Matrix Exponential

 Exercises

6 Symmetry

 6.1 Symmetry of Plane Figures

 6.2 Isometries

 6.3 Isometries of the Plane

 6.4 Finite Groups of Orthogonal Operators on the Plane

 6.5 Discrete Groups of Isometries

 6.6 Plane Crystallographic Gloups

 6.7 Abstract Symmetry: Group Operations

 6.8 The Operation on Cosets

 6.9 The Counting Formula

 6.10 Operations on Subsets

 6.11 Permutation Representations

 6.12 Finite Subgroups cf the Rotation Group

 Exercises

7 More Group Theory

 7.1 Cayley's Theorem

 7.2 The Class Equation

 7.3 p-Groups

 7.4 The Class Equation of the Icosahedral Group

 7.5 Conjugation in the Symmetric Group

 7.6 Normalizers

 7.7 The Sylow Theorems

 7.8 Groups of Order 12

 7.9 The Free Group

 7.10 Generators and Relations

 7.11 The Todd-Coxeter Algorithm

 Exercises

8 Bilinear Forms

 8.1 Bilinear Forms

 8.2 Symmetric Forms

 8.3 Hermitian Forms

 8.4 Orthogonality

 8.5 Euclidean Spaces and Hermitian Spaces

 8.6 The Spectral Theorem

 8.7 Conics and Quadrics

 8.8 Skew-Symmetric Forms

 8.9 Summary

 Exercises

9 Linear Groups

 9.1 The Classical Groups

 9.2 Interlude: Spheres

 9.3 The Special Unitary Group SU2

 9.4 The Rotation Group S03

 9.5 One-Parameter Groups

 9.6 The Lie Algebra

 9.7 Translation in a Group

 9.8 Normal Subgroups of SL2

 Exercises

10 Group Representations

 10.1 Definitions

 10.2 Irreducible Representations

 10.3 Unitary Representations

 10.4 Characters

 10.5 One-Dimensional Characters

 10.6 The Regular Representation

 10.7 Schur's Lemma

 10.8 Proof of the Orthogonality Relations .

 10.9 Representations of SU2

 Exercises

11 Rings

 11.1 Definition of a Ring

 11.2 Polynomial Rings

 11.3 Homomorphisms and Ideals

 11.4 Quotient Rings

 11.5 Adjoining Elements

 11.6 Product Rings

 11.7 Fractions

 11.8 Maximal Ideals

 11.9 Algebraic Geometry

 Exercises

12 Factoring

 12.1 Factoring Integers

 12.2 Unique Factorization Domains

 12.3 Gauss's Lemma

 12.4 Factoring Integer Polynomials

 12.5 Gauss Primes

 Exercises

13 Quadratic Number Fields

 13.1 Algebraic Integers

 13.2 Factoring Algebraic Integers

 13.3 Ideals in Z[□]

 13.4 Ideal Multiplication

 13.5 Factoring Ideals

 13.6 Prime Ideals and Prime Integers

 13.7 Ideal Classes

 13.8 Computing the Class Group

 13.9 Real Quadratic Fields

 13.10 About Lattices

 Exercises

14 Linear Algebra in a Ring

 14.1 Modules

 14.2 Free Modules

 14.3 Identities

 14.4 Diagonalizing Integer Matrices

 14.5 Generators and Relations

 14.6 Noetherian Rings

 14.7 Structure of Abelian Groups

 14.8 Application to Linear Operators

 14.9 Polynomial Rings in Several Variables

 Exercises

15 Fields

 15.1 Examples of Fields

 15.2 Algebraic and Transcendental Elements

 15.3 The Degree of a Field Extension

 15.4 Finding the Irreducible Polynomial

 15.5 Ruler and Compass Constructions

 15.6 Adjoining Roots

 15.7 Finite Fields

 15.8 Primitive Elements

 15.9 Function Fields

 15.10 The Fundamental Theorem of Algebra

 Exercises

16 Galois Theory

 16.1 Symmetric Functions

 16.2 The Discriminant

 16.3 Splitting Fields

 16.4 Isomorphisms of Field Extensions

 16.5 Fixed Fields

 16.6 Galois Extensions

 16.7 The Main Theorem

 16.8 Cubic Equations

 16.9 Quartic Equations

 16.10 Roots of Unity

 16.11 Kummer Extensions

 16.12 QuinticEquations

 Exercises

APPENDIX

 Background Material

 A.1 About Proofs

 A.2 The Integers

 A.3 Zorn's Lemma

 A.4 The Implicit Function Theorem

 Exercises

Bibliography

Notation

Index