基础代数

书籍目录

目录

Preface

Conventions on Terminology

1.Sets

1.1 Finite， Countable and Uncountable Sets

1，2 Zom's Lemma and Well—ordered Sets

1.3 Graphs

2.Groups

2.1 Definition and Basic Properties

2.2 Permutation Groups

2.3 The Isomorphism Theorems

2.4 Soluble and Nilpotent Groups

2.5 Commutators

2.6 The Frattini Subgroup and the Fitting Subgroup

3.Lattices and Categories

3.1 Definitions； Modular and Distributive Lattices

3.2 Chain Conditions

3.3 Categories

3.4 Boolean Algebras

4.Rings and Modules

4.1 The Definitions Recalled

4.2 The Category of Modules over a Ring

4.3 Semisimple Modules

4.4 Matrix Rings

4.5 Direct Products of Rings

4.6 Free Modules

4.7 Projective and Injective Modules

4.8 The Tensor Product of Modules

4.9 Duality of Finite Abelian Groups

5.Algebras

5.1 Algebras； Definition and Examples

5.2 The Wedderbum Structure Theorems

5.3 The Radical

5.4 The Tensor Product of Algebras

5.5 The Regular Representation； Norm and Trace

5.6 M6bius Functions

6.Muhilinear Algebra

6.1 Graded Algebras

6.2 Free Algebras and Tensor Algebras

6.3 The Hilbert Series of a Graded Ring or Module

6.4 The Exterior Algebra on a Module

7.Field Theory

7.1 Fields and their Extensions

7.2 Splitting Fields

7.3 The Algebraic Closure of a Field

7.4 Separability

7.5 Automorphisms of Field Extensions

7.6 The Fundamental Theorem of Galois Theory

7.7 Roots of Unity

7.8 Finite Fields

7.9 Primitive Elements； Norm and Trace

7.10 Galois Theory of Equations

7.11 The Solution of Equations by Radicals

8.Quadratic Forms and Ordered Fields

8.1 Inner Product Spaces

8.2 Orthogonal Sums and Diagonalization

8.3 The Orthogonal Group of a Space

8.4 The Clifford Algebra and the Spinor Norm

8.5 Witt's Cancellation Theorem and the Witt Group of a Field

8.6 Ordered Fields

8.7 The Field of Real Numbers

8.8 Formally Real Fields

8.9 The Witt Ring of a Field

8.10 The Symplectic Group

8.11 Quadratic Forms in Characteristic Two

9.Valuation Theory

9.1 Divisibility and Valuations

9.2 Absolute Values

9.3 The p—adic Numbers

9.4 Integral Elements

9.5 Extension of Valuations

10.Commutative Rings

10.1 Operations on Ideals

10.2 Prime Ideals and Factorization

10.3 Localization

10.4 Noetherian Rings

10.5 Dedekind Domains

10.6 Modules over Dedekind Domains

10.7 Algebraic Equations

10.8 The Primary Decomposition

10.9 Dimension

10.10 The Hilbert Nullstellensatz

11.Infinite Field Extensions

11.1 Abstract Dependence Relations

11.2 Algebraic Dependence

11.3 Simple Transcendental Extensions

11.4 Separable and p—radical Extensions

11.5 Derivations

11.6 Linearly Disjoint Extensions

11.7 Composites of Fields

11.8 Infinite Algebraic Extensions

11.9 Galois Descent

11.10 Kummer Extensions

Bibliography

List of Notations

Author Index

Subject Index

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