基础代数

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