近世代数也称抽象代数，是现代数学的重要基础，主要研究群、环、域等代数结构。本书出自抽象代数领域的两位巨匠之手，曾对近世代数教学产生深远的影响，帮助了几代学子理解和掌握近世代数，至今本书仍是一部对自学和课堂教学都极具价值的参考书和教材。作者用大家热悉且具体的例子来阐述每一个概念，深入浅出，透彻简洁。为了培养学生独立思考的能力，每个专题都包括丰富的练习。

本书出自近世代数领域的两位科学巨匠之手，是一本经典的教材。全书共分为15章，内容包括：整数、多项式、实数、复数、矩阵代数、线性群、行列式和标准型、布尔代数和格、超限算术、环和理想、代数数域和伽罗华理论等。

本书曾帮助过几代人理解近世代数，至今仍是一本非常有价值的参考书和教材，适合数学专业及其他理工科专业高年级本科生和研究生使用。

Preface to the Fourth Edition

1　The Integers

 1.1　Commutative Rings; Integral Domains

 1.2　Elementary Properties of Commutative Rings

 1.3　Ordered Domains

 1.4　Well-Ordering Principle

 1.5　Finite Induction; Laws of Exponents

 1.6　ivisibility

 1.7　The Euclidean Algorithm

 1.8　Fundamental Theorem of Arithmetic

 1.9　Congruences

 1.10　The Rings Zn

 1.11　Sets, Functions, and Relations

 1.12　Isomorphisms and Automorphisms

2　Rational Numbers and Fields

 2.1　Definition of a Field

 2.2　Construction of the Rationals

 2.3　Simultaneous Linear Equations

 2.4　Ordered Fields

 2.5　Postulates for the Positive Integers

 2.6　Peano Postulates

3　Polynomials

 3.1　Polynomial Forms

 3.2　Polynomial Functions

 3.3　Homomorphisms of Commutative Rings

 3.4　Polynomials in Several Variables

 3.5　The Division Algorithm

 3.6　Units and Associates

 3.7　Irreducible Polynomials

 3.8　Unique Factorization Theorem

 3.9　Other Domains with Unique Factorization

 3.10　Eisenstein's Irreducibility Criterion

 3.11　Partial Fractions

4　Real Numbers

 4.1　Dilemma of Pythagoras

 4.2　Upper and Lower Bounds

 4.3　Postulates for Real Numbers

 4.4　Roots of Polynomial Equations

 4.5　Dedekind Cuts

5　Complex Numbers

 5.1　Definition

 5.2　The Complex Plane

 5.3　Fundamental Theorem of Algebra

 5.4　Conjugate Numbers and Real Polynomials

 5.5　Quadratic and Cubic Equations

 5.6　Solution of Quartic by Radicals

 5.7　Equations of Stable Type

6　Groups

 6.1　Symmetries of the Square

 6.2　Groups of Transformations

 6.3　Further Examples

 6.4　Abstract Groups

 6.5　Isomorphism

 6.6　Cyclic Groups

 6.7　Subgroups

 6.8　Lagrange's Theorem

 6.9　Permutation Groups

 6.10　Even and Odd Permutations

 6.11　Homomorphisms

 6.12　Automorphisms; Conjugate Elements

 6.13　Quotient Groups

 6.14　Equivalence and Congruence Relations

7　Vectors and Vector Spaces

 7.1　Vectors in a Plane

 7.2　Generalizations

 7.3　Vector Spaces and Subspaces

 7.4　Linear Independence and Dimension

 7.5　Matrices and Row-equivalence

 7.6　Tests for Linear Dependence

 7.7　Vector Equations; Homogeneous Equations

 7.8　Bases and Coordinate Systems

 7.9　Inner Products

 7.10　Euclidean Vector Spaces

 7.11　Normal Orthogonal Bases

 7.12　Quotient-spaces

 7.13　Linear Functions and Dual Spaces

8　The Algebra of Matrices

 8.1　Linear Transformations and Matrices

 8.2　Matrix Addition

 8.3　Matrix Multiplication

 8.4　Diagonal, Permutation, and Triangular Matrices

 8.5　Rectangular Matrices

 8.6　Inverses

 8.7　Rank and Nullity

 8.8　Elementary Matrices

 8.9　Equivalence and Canonical Form

 8.10　Bilinear Functions and Tensor Products

 8.11　Quaternions

9　Linear Groups

 9.1　Change of Basis

 9.2　Similar Matrices and Eigenvectors

 9.3　The Full Linear and Affine Groups

 9.4　The Orthogonal and Euclidean Groups

 9.5　Invariants and Canonical Forms

 9.6　Linear and Bilinear Forms

 9.7　Quadratic Forms

 9.8　Quadratic Forms Under the Full Linear Group

 9.9　Real Quadratic Forms Under the Full Linear Group

 9.10　Quadratic Forms Under the Orthogonal Group

 9.11　Quadrics Under the Affine and Euclidean Groups

 9.12　Unitary and Hermitian Matrices

 9.13　Affine Geometry

 9.14　Projective Geometry

10　Determinants and Canonical Forms

 10.1　Definition and Elementary Properties of Determinants

 10.2　Products of Determinants

 10.3　Determinants as Volumes

 10.4　The Characteristic Polynomial

 10.5　The Minimal Polynomial

 10.6　Cayley-Hamilton Theorem

 10.7　Invariant Subspaces and Reducibility

 10.8　First Decomposition Theorem

 10.9　Second Decomposition Theorem

 10.10　Rational and Jordan Canonical Forms

11　Boolean Algebras and Lattices

 11.1　Basic Definition

 11.2　Laws: Analogy with Arithmetic

 11.3　Boolean Algebra

 11.4　Deduction of Other Basic Laws

 11.5　Canonical Forms of Boolean Polynomials

 11.6　Partial Orderings

 11.7　Lattices

 11.8　Representation by Sets

12　Transfinite Arithmetic

 12.1　Numbers and Sets

 12.2　Countable Sets

 12.3　Other Cardinal Numbers

 12.4　Addition and Multiplication of Cardinals

 12.5　Exponentiation

13　Rings and Ideals

 13.1　Rings

 13.2　Homomorphisms

 13.3　Quotient-rings

 13.4　Algebra of Ideals

 13.5　Polynomial Ideals

 13.6　Ideals in Linear Algebras

 13.7　The Characteristic of a Ring

 13.8　Characteristics of Fields

14　Algebraic Number Fields

 14.1　Algebraic and Transcendental Extensions

 14.2　Elements Algebraic over a Field

 14.3　Adjunction of Roots

 14.4　Degrees and Finite Extensions

 14.5　Iterated Algebraic Extensions

 14.6　Algebraic Numbers

 14.7　Gaussian Integers

 14.8　Algebraic Integers

 14.9　Sums and Products of Integers

 14.10　Factorization of Quadratic Integers

15　Galois Theory

 15.1　Root Fields for Equations

 15.2　Uniqueness Theorem

 15.3　Finite Fields

 15.4　The Galois Group

 15.5　Separable and Inseparable Polynomials

 15.6　Properties of the Galois Group

 15.7　Subgroups and Subfields

 15.8　Irreducible Cubic Equations

 15.9　Insolvability of Quintic Equations

Bibliography

List of Special Symbols

Index