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代数(英文版第2版)/华章数学原版精品系列

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作者(美)阿廷

出版社机械工业

ISBN9787111367017

出版时间2012-01

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货号2169318

上书时间2024-11-04

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商品描述: 导语摘要
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。

作者简介
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。

目录
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

内容摘要
《代数(英文版)(第2版)》由著名代数学家与代数几何学家MichaelArtin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。

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