基础代数

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