正版 代数(英文版第2版)/华章数学原版精品系列 (美)阿廷 机械工

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

阿廷（MichaelArtin），当代领袖型代数学家与代数几何学家之一。美国麻省理工学院数学系荣誉退休教授。1990年至1992年。曾担任美国数学学会主席。由于他在交换代数与非交换代数、环论以及现代代数几何学等方面做出的贡献，2002年获得美国数学学会颁发的LeroyP.Steele终身成就奖。Artin的主要贡献包括他的逼近定理、在解决沙法列维奇-泰特猜测中的工作以及为推广“概形”而创建的“代数空间”概念。

著名代数学家与代数几何学家MichaelArtin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。