作者写这本书的一个最重要的原因是想给读者一个系统的、综合的、完整的描述数字体系的理论，从而形成一个基础结构，在数学的各个分支中发挥中心作用。写这本书的目标是将一个数论和代数的入门本科课程发展为一个综合的学科。
这本书由10章组成，从集合论的元素开始（第1章），第2章是关于矩阵和行列式的，第3、4、5、6章涵盖了与线性代数相关的主要内容。我们在第3章中介绍了一些域论的元素，这些元素是描述线性代数所需要的一些基本元素（如矢量空间和双线性形式），不仅在数域上，而且在有限域上都是不可缺少的。第3章、第6章、第7章和第8章阐述了代数结构的主要思想。而第9章和第10章则展示了代数思想在数论中的应用（如9。4节）。第10章是对实数系统及其主要子系统的严格构造的发展，这一章是本书主要内容的一个重要附录，有无法忽略的重要价值。

PREFACE
CHAPTER 1 SETS
1.1 Operations on Sets l
Exercise Set 1.1
1.2 Set Mappings
Exercise Set 1.2
1.3 Products of Mappings
Exercise Set 1.3
1.4 Some Properties of Integers
Exercise Set 1.4
CHAPTER 2 MATRICES AND DETERMINANTS
2.1 Operations on Matrices
Exercise Set 2.1
2.2 Permutations of Finite Sets
Exercise Set 2.2
2.3 Determinants of Matrices
Exercise Set 2.3
2.4 Computing Determinants
Exercise Set 2.4
2.5 Properties of the Product of Matrices
Exercise Set 2.5
CHAPTER 3 FIELDS
3.1 Binary Algebraic Operations
Exercise Set 3.1
3.2 Basic Properties of Fields
Exercise Set 3.2
3.3 The Field of Complex Numbers
Exercise Set 3.3
CHAPTER 4 VECTOR SPACES
4.1 Vector Spaces
Exercise Set 4.1
4.2 Dimension
Exercise Set 4.2
4.3 The Rank of a Matrix
Exercise Set 4.3
4.4 Quotient Spaces
Exercise Set 4.4
CHAPTER 5 LINEAR MAPPINGS
5.1 Linear Mappings
Exercise Set 5.1
5.2 Matrices of Linear Mappings
Exercise Set 5.2
5.3 Systems of Linear Equations
Exercise Set 5.3
5,4 Eigenvectors and Eigenvalues
Exercise Set 5.4
CHAPTER 6 BILINEAR FORMS
6.1 Bilinear Forms
Exercise Set 6.1
6.2 Classical Forms
Exercise Set 6,2
6.3 Symmetric Forms over R
Exercise Set 6,3
6.4 Euclidean Spaces
Exercise Set 6.4
CHAPTER 7 RINGS
7.1 Rings, Subrings, and Examples
Exercise Set 7.1
7.2 Equivalence Relations
Exercise Set 7.2
7.3 Ideals and Quotient Rings
Exercise Set 7.3
7.4 Homomorphisms of Rings
Exercise Set 7.4
7.5 Rings of Polynomials and Formal Power Series
Exercise Set 7.5
7.6 Rings of Multivariable Polynomials
Exercise Set 7.6
CHAPTER 8 GROUPS
8.1 Groups and Subgroups
Exercise Set 8.1
8.2 Examples of Groups and Subgroups
Exercise Set 8.2
8.3 Cosets
Exercise Set 8.3
8.4 Normal Subgroups and Factor Groups
Exercise Set 8.4
8.5 Homomorphisms of Groups
Exercise Set 8.5
CHAPTER 9 ARITHMETIC PROPERTIES OF RINGS
9.1 Extending Arithmetic to Commutative Rings
Exercise Set 9.1
9.2 Euclidean Rings
Exercise Set 9.2
9.3 Irreducible Polynomials
Exercise Set 9.3
9.4 Arithmetic Functions
Exercise Set 9.4
9.5 Congruences
Exercise Set 9.5
CHAPTER 10 THE REAL NUMBER SYSTEM
10.1 The Natural Numbers
10.2 The Integers
10.3 The Rationals
10.4 The Real Numbers
ANSWERS TO SELECTED EXERCISES
INDEX
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