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群、环、域导引（影印版）

样章
作者：
Fernando Q. Gouvêa
定价：
135.00 元
版面字数：
560千字
开本：
特殊
装帧形式：
精装
版次：
1
最新版次
印刷时间：
2022年
ISBN：
978-7-04-056999-5
物料号：
56999-00
出版时间：
2022-02-28
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
代数学
暂无
  • 目录
    • 前辅文
      • A Guide to this Guide
        • 1 Algebra: Classical,Modern, and Ultramodern
          • 1.1 The Beginnings of Modern Algebra
            • 1.2 Modern Algebra
              • 1.3 Ultramodern Algebra
                • 1.4 What Next?
                • 2 Categories
                  • 2.1 Categories
                    • 2.2 Functors
                      • 2.3 Natural Transformations
                        • 2.4 Products, Coproducts, and Generalizations
                        • 3 Algebraic Structures
                          • 3.1 Structures with One Operation
                            • 3.2 Rings
                              • 3.3 Actions
                                • 3.4 Semirings
                                  • 3.5 Algebras
                                    • 3.6 Ordered Structures
                                    • 4 Groups and their Representations
                                      • 4.1 Definitions
                                        • 4.1.1 Groups and homomorphisms
                                          • 4.1.2 Subgroups
                                            • 4.1.3 Actions
                                              • 4.1.4 G acting on itself
                                              • 4.2 Some Important Examples
                                                • 4.2.1 Permutation groups
                                                  • 4.2.2 Symmetry groups
                                                    • 4.2.3 Other examples
                                                      • 4.2.4 Topological groups
                                                        • 4.2.5 Free groups
                                                        • 4.3 Reframing the Definitions
                                                          • 4.4 Orbits and Stabilizers
                                                            • 4.4.1 Stabilizers
                                                              • 4.4.2 Orbits
                                                                • 4.4.3 Acting by multiplication
                                                                  • 4.4.4 Cosets
                                                                    • 4.4.5 Counting cosets and elements
                                                                      • 4.4.6 Double cosets
                                                                        • 4.4.7 A nice example
                                                                        • 4.5 Homomorphisms and Subgroups
                                                                          • 4.5.1 Kernel, image, quotient
                                                                            • 4.5.2 Homomorphism theorems
                                                                              • 4.5.3 Exact sequences
                                                                                • 4.5.4 H¨older’s dream
                                                                                • 4.6 Many Cheerful Subgroups
                                                                                  • 4.6.1 Generators, cyclic groups
                                                                                    • 4.6.2 Elements of finite order
                                                                                      • 4.6.3 Finitely generated groups and the Burnside problem
                                                                                        • 4.6.4 Other nice subgroups
                                                                                          • 4.6.5 Conjugation and the class equation
                                                                                            • 4.6.6 p-groups
                                                                                              • 4.6.7 Sylow’s Theorem and Sylow subgroups
                                                                                              • 4.7 Sequences of Subgroups
                                                                                                • 4.7.1 Composition series
                                                                                                  • 4.7.2 Central series, derived series, nilpotent, solvable
                                                                                                  • 4.8 New Groups from Old
                                                                                                    • 4.8.1 Direct products
                                                                                                      • 4.8.2 Semidirect products
                                                                                                        • 4.8.3 Isometries of R3
                                                                                                          • 4.8.4 Free products
                                                                                                            • 4.8.5 Direct sums of abelian groups
                                                                                                              • 4.8.6 Inverse limits and direct limits
                                                                                                              • 4.9 Generators and Relations
                                                                                                                • 4.9.1 Definition and examples
                                                                                                                  • 4.9.2 Cayley graphs
                                                                                                                    • 4.9.3 The word problem
                                                                                                                    • 4.10 Abelian Groups
                                                                                                                      • 4.10.1 Torsion
                                                                                                                        • 4.10.2 The structure theorem
                                                                                                                        • 4.11 Small Groups
                                                                                                                          • 4.11.1 Order four, order p2
                                                                                                                            • 4.11.2 Order six, order pq
                                                                                                                              • 4.11.3 Order eight, order p3
                                                                                                                                • 4.11.4 And so on
                                                                                                                                • 4.12 Groups of Permutations
                                                                                                                                  • 4.12.1 Cycle notation and cycle structure
                                                                                                                                    • 4.12.2 Conjugation and cycle structure
                                                                                                                                      • 4.12.3 Transpositions as generators
                                                                                                                                        • 4.12.4 Signs and the alternating groups
                                                                                                                                          • 4.12.5 Transitive subgroups
                                                                                                                                            • 4.12.6 Automorphismgroup of Sn
                                                                                                                                            • 4.13 Some Linear Groups
                                                                                                                                              • 4.13.1 Definitions and examples
                                                                                                                                                • 4.13.2 Generators
                                                                                                                                                  • 4.13.3 The regular representation
                                                                                                                                                    • 4.13.4 Diagonal and upper triangular
                                                                                                                                                      • 4.13.5 Normal subgroups
                                                                                                                                                        • 4.13.6 PGL
                                                                                                                                                          • 4.13.7 Linear groups over finite fields
                                                                                                                                                          • 4.14 Representations of Finite Groups
                                                                                                                                                            • 4.14.1 Definitions
                                                                                                                                                              • 4.14.2 Examples
                                                                                                                                                                • 4.14.3 Constructions
                                                                                                                                                                  • 4.14.4 Decomposing into irreducibles
                                                                                                                                                                    • 4.14.5 Direct products
                                                                                                                                                                      • 4.14.6 Characters
                                                                                                                                                                        • 4.14.7 Character tables
                                                                                                                                                                          • 4.14.8 Going through quotients
                                                                                                                                                                            • 4.14.9 Going up and down
                                                                                                                                                                              • 4.14.10 Representations of S4
                                                                                                                                                                            • 5 Rings and Modules
                                                                                                                                                                              • 5.1 Definitions
                                                                                                                                                                                • 5.1.1 Rings
                                                                                                                                                                                  • 5.1.2 Modules
                                                                                                                                                                                    • 5.1.3 Torsion
                                                                                                                                                                                      • 5.1.4 Bimodules
                                                                                                                                                                                        • 5.1.5 Ideals
                                                                                                                                                                                          • 5.1.6 Restriction of scalars
                                                                                                                                                                                            • 5.1.7 Rings with few ideals
                                                                                                                                                                                            • 5.2 More Examples, More Definitions
                                                                                                                                                                                              • 5.2.1 The integers
                                                                                                                                                                                                • 5.2.2 Fields and skew fields
                                                                                                                                                                                                  • 5.2.3 Polynomials
                                                                                                                                                                                                    • 5.2.4 R-algebras
                                                                                                                                                                                                      • 5.2.5 Matrix rings
                                                                                                                                                                                                        • 5.2.6 Group algebras
                                                                                                                                                                                                          • 5.2.7 Monsters
                                                                                                                                                                                                            • 5.2.8 Some examples of modules
                                                                                                                                                                                                              • 5.2.9 Nil and nilpotent ideals
                                                                                                                                                                                                                • 5.2.10 Vector spaces as KOEX-modules
                                                                                                                                                                                                                  • 5.2.11 Q and Q/Z as Z-modules
                                                                                                                                                                                                                    • 5.2.12 Why study modules?
                                                                                                                                                                                                                    • 5.3 Homomorphisms, Submodules, and Ideals
                                                                                                                                                                                                                      • 5.3.1 Submodules and quotients
                                                                                                                                                                                                                        • 5.3.2 Quotient rings
                                                                                                                                                                                                                          • 5.3.3 Irreducible modules, simple rings
                                                                                                                                                                                                                            • 5.3.4 Small and large submodules
                                                                                                                                                                                                                            • 5.4 Composing and Decomposing
                                                                                                                                                                                                                              • 5.4.1 Direct sums and products
                                                                                                                                                                                                                                • 5.4.2 Complements
                                                                                                                                                                                                                                  • 5.4.3 Direct and inverse limits
                                                                                                                                                                                                                                    • 5.4.4 Products of rings
                                                                                                                                                                                                                                    • 5.5 Free Modules
                                                                                                                                                                                                                                      • 5.5.1 Definitions and examples
                                                                                                                                                                                                                                        • 5.5.2 Vector spaces
                                                                                                                                                                                                                                          • 5.5.3 Traps
                                                                                                                                                                                                                                            • 5.5.4 Generators and free modules
                                                                                                                                                                                                                                              • 5.5.5 Homomorphisms of free modules
                                                                                                                                                                                                                                                • 5.5.6 Invariant basis number
                                                                                                                                                                                                                                                • 5.6 Commutative Rings, Take One
                                                                                                                                                                                                                                                  • 5.6.1 Prime ideals
                                                                                                                                                                                                                                                    • 5.6.2 Primary ideals
                                                                                                                                                                                                                                                      • 5.6.3 The Chinese Remainder Theorem
                                                                                                                                                                                                                                                      • 5.7 Rings of Polynomials
                                                                                                                                                                                                                                                        • 5.7.1 Degree
                                                                                                                                                                                                                                                          • 5.7.2 The evaluation homomorphism
                                                                                                                                                                                                                                                            • 5.7.3 Integrality
                                                                                                                                                                                                                                                              • 5.7.4 Unique factorization and ideals
                                                                                                                                                                                                                                                                • 5.7.5 Derivatives
                                                                                                                                                                                                                                                                  • 5.7.6 Symmetric polynomials and functions
                                                                                                                                                                                                                                                                    • 5.7.7 Polynomials as functions
                                                                                                                                                                                                                                                                    • 5.8 The Radical, Local Rings, and Nakayama’s Lemma
                                                                                                                                                                                                                                                                      • 5.8.1 The Jacobson radical
                                                                                                                                                                                                                                                                        • 5.8.2 Local rings
                                                                                                                                                                                                                                                                          • 5.8.3 Nakayama’s Lemma
                                                                                                                                                                                                                                                                          • 5.9 Commutative Rings: Localization
                                                                                                                                                                                                                                                                            • 5.9.1 Localization
                                                                                                                                                                                                                                                                              • 5.9.2 Fields of fractions
                                                                                                                                                                                                                                                                                • 5.9.3 An important example
                                                                                                                                                                                                                                                                                  • 5.9.4 Modules under localization
                                                                                                                                                                                                                                                                                    • 5.9.5 Ideals under localization
                                                                                                                                                                                                                                                                                      • 5.9.6 Integrality under localization
                                                                                                                                                                                                                                                                                        • 5.9.7 Localization at a prime ideal
                                                                                                                                                                                                                                                                                          • 5.9.8 What if R is not commutative?
                                                                                                                                                                                                                                                                                          • 5.10 Hom
                                                                                                                                                                                                                                                                                            • 5.10.1 Making Hom a module
                                                                                                                                                                                                                                                                                              • 5.10.2 Functoriality
                                                                                                                                                                                                                                                                                                • 5.10.3 Additivity
                                                                                                                                                                                                                                                                                                  • 5.10.4 Exactness
                                                                                                                                                                                                                                                                                                  • 5.11 Tensor Products
                                                                                                                                                                                                                                                                                                    • 5.11.1 Definition and examples
                                                                                                                                                                                                                                                                                                      • 5.11.2 Examples
                                                                                                                                                                                                                                                                                                        • 5.11.3 Additivity and exactness
                                                                                                                                                                                                                                                                                                          • 5.11.4 Over which ring?
                                                                                                                                                                                                                                                                                                            • 5.11.5 When R is commutative
                                                                                                                                                                                                                                                                                                              • 5.11.6 Extension of scalars, aka base change
                                                                                                                                                                                                                                                                                                                • 5.11.7 Extension and restriction
                                                                                                                                                                                                                                                                                                                  • 5.11.8 Tensor products and Hom
                                                                                                                                                                                                                                                                                                                    • 5.11.9 Finite free modules
                                                                                                                                                                                                                                                                                                                      • 5.11.10 Tensoring a module with itself
                                                                                                                                                                                                                                                                                                                        • 5.11.11 Tensoring two rings
                                                                                                                                                                                                                                                                                                                        • 5.12 Projective, Injective, Flat
                                                                                                                                                                                                                                                                                                                          • 5.12.1 Projective modules
                                                                                                                                                                                                                                                                                                                            • 5.12.2 Injective modules
                                                                                                                                                                                                                                                                                                                              • 5.12.3 Flat modules
                                                                                                                                                                                                                                                                                                                                • 5.12.4 If R is commutative
                                                                                                                                                                                                                                                                                                                                • 5.13 Finiteness Conditions for Modules
                                                                                                                                                                                                                                                                                                                                  • 5.13.1 Finitely generated and finitely cogenerated
                                                                                                                                                                                                                                                                                                                                    • 5.13.2 Artinian and Noetherian
                                                                                                                                                                                                                                                                                                                                      • 5.13.3 Finite length
                                                                                                                                                                                                                                                                                                                                      • 5.14 Semisimple Modules
                                                                                                                                                                                                                                                                                                                                        • 5.14.1 Definitions
                                                                                                                                                                                                                                                                                                                                          • 5.14.2 Basic properties
                                                                                                                                                                                                                                                                                                                                            • 5.14.3 Socle and radical
                                                                                                                                                                                                                                                                                                                                              • 5.14.4 Finiteness conditions
                                                                                                                                                                                                                                                                                                                                              • 5.15 Structure of Rings
                                                                                                                                                                                                                                                                                                                                                • 5.15.1 Finiteness conditions for rings
                                                                                                                                                                                                                                                                                                                                                  • 5.15.2 Simple Artinian rings
                                                                                                                                                                                                                                                                                                                                                    • 5.15.3 Semisimple rings
                                                                                                                                                                                                                                                                                                                                                      • 5.15.4 Artinian rings
                                                                                                                                                                                                                                                                                                                                                        • 5.15.5 Non-Artinian rings
                                                                                                                                                                                                                                                                                                                                                        • 5.16 Factorization in Domains
                                                                                                                                                                                                                                                                                                                                                          • 5.16.1 Units, irreducibles, and the rest
                                                                                                                                                                                                                                                                                                                                                            • 5.16.2 Existence of factorization
                                                                                                                                                                                                                                                                                                                                                              • 5.16.3 Uniqueness of factorization
                                                                                                                                                                                                                                                                                                                                                                • 5.16.4 Principal ideal domains
                                                                                                                                                                                                                                                                                                                                                                  • 5.16.5 Euclidean domains
                                                                                                                                                                                                                                                                                                                                                                    • 5.16.6 Greatest common divisor
                                                                                                                                                                                                                                                                                                                                                                      • 5.16.7 Dedekind domains
                                                                                                                                                                                                                                                                                                                                                                      • 5.17 Finitely Generated Modules over Dedekind Domains
                                                                                                                                                                                                                                                                                                                                                                        • 5.17.1 The structure theorems
                                                                                                                                                                                                                                                                                                                                                                          • 5.17.2 Application to abelian groups
                                                                                                                                                                                                                                                                                                                                                                            • 5.17.3 Application to linear transformations
                                                                                                                                                                                                                                                                                                                                                                          • 6 Fields and Skew Fields
                                                                                                                                                                                                                                                                                                                                                                            • 6.1 Fields and Algebras
                                                                                                                                                                                                                                                                                                                                                                              • 6.1.1 Some examples
                                                                                                                                                                                                                                                                                                                                                                                • 6.1.2 Characteristic and prime fields
                                                                                                                                                                                                                                                                                                                                                                                  • 6.1.3 K-algebras and extensions
                                                                                                                                                                                                                                                                                                                                                                                    • 6.1.4 Two kinds of K-homomorphisms
                                                                                                                                                                                                                                                                                                                                                                                      • 6.1.5 Generating sets
                                                                                                                                                                                                                                                                                                                                                                                        • 6.1.6 Compositum
                                                                                                                                                                                                                                                                                                                                                                                          • 6.1.7 Linear disjointness
                                                                                                                                                                                                                                                                                                                                                                                          • 6.2 Algebraic Extensions
                                                                                                                                                                                                                                                                                                                                                                                            • 6.2.1 Definitions
                                                                                                                                                                                                                                                                                                                                                                                              • 6.2.2 Transitivity
                                                                                                                                                                                                                                                                                                                                                                                                • 6.2.3 Working without an A
                                                                                                                                                                                                                                                                                                                                                                                                  • 6.2.4 Norm and trace
                                                                                                                                                                                                                                                                                                                                                                                                    • 6.2.5 Algebraic elements and homomorphisms
                                                                                                                                                                                                                                                                                                                                                                                                      • 6.2.6 Splitting fields
                                                                                                                                                                                                                                                                                                                                                                                                        • 6.2.7 Algebraic closure
                                                                                                                                                                                                                                                                                                                                                                                                        • 6.3 Finite Fields
                                                                                                                                                                                                                                                                                                                                                                                                          • 6.4 Transcendental Extensions
                                                                                                                                                                                                                                                                                                                                                                                                            • 6.4.1 Transcendence basis
                                                                                                                                                                                                                                                                                                                                                                                                              • 6.4.2 Geometric examples
                                                                                                                                                                                                                                                                                                                                                                                                                • 6.4.3 Noether Normalization
                                                                                                                                                                                                                                                                                                                                                                                                                  • 6.4.4 Luroth’s Theorem
                                                                                                                                                                                                                                                                                                                                                                                                                    • 6.4.5 Symmetric functions
                                                                                                                                                                                                                                                                                                                                                                                                                    • 6.5 Separability
                                                                                                                                                                                                                                                                                                                                                                                                                      • 6.5.1 Separable and inseparable polynomials
                                                                                                                                                                                                                                                                                                                                                                                                                        • 6.5.2 Separable extensions
                                                                                                                                                                                                                                                                                                                                                                                                                          • 6.5.3 Separability and tensor products
                                                                                                                                                                                                                                                                                                                                                                                                                            • 6.5.4 Norm and trace
                                                                                                                                                                                                                                                                                                                                                                                                                              • 6.5.5 Purely inseparable extensions
                                                                                                                                                                                                                                                                                                                                                                                                                                • 6.5.6 Separable closure
                                                                                                                                                                                                                                                                                                                                                                                                                                  • 6.5.7 Primitive elements
                                                                                                                                                                                                                                                                                                                                                                                                                                  • 6.6 Automorphisms and Normal Extensions
                                                                                                                                                                                                                                                                                                                                                                                                                                    • 6.6.1 Automorphisms
                                                                                                                                                                                                                                                                                                                                                                                                                                      • 6.6.2 Normal extensions
                                                                                                                                                                                                                                                                                                                                                                                                                                      • 6.7 Galois Theory
                                                                                                                                                                                                                                                                                                                                                                                                                                        • 6.7.1 Galois extensions and Galois groups
                                                                                                                                                                                                                                                                                                                                                                                                                                          • 6.7.2 The Galois group as topological group
                                                                                                                                                                                                                                                                                                                                                                                                                                            • 6.7.3 The Galois correspondence
                                                                                                                                                                                                                                                                                                                                                                                                                                              • 6.7.4 Composita
                                                                                                                                                                                                                                                                                                                                                                                                                                                • 6.7.5 Norm and trace
                                                                                                                                                                                                                                                                                                                                                                                                                                                  • 6.7.6 Normal bases
                                                                                                                                                                                                                                                                                                                                                                                                                                                    • 6.7.7 Solution by radicals
                                                                                                                                                                                                                                                                                                                                                                                                                                                      • 6.7.8 Determining Galois groups
                                                                                                                                                                                                                                                                                                                                                                                                                                                        • 6.7.9 The inverse Galois problem
                                                                                                                                                                                                                                                                                                                                                                                                                                                          • 6.7.10 Analogies and generalizations
                                                                                                                                                                                                                                                                                                                                                                                                                                                          • 6.8 Skew Fields and Central Simple Algebras
                                                                                                                                                                                                                                                                                                                                                                                                                                                            • 6.8.1 Definition and basic results
                                                                                                                                                                                                                                                                                                                                                                                                                                                              • 6.8.2 Quaternion algebras
                                                                                                                                                                                                                                                                                                                                                                                                                                                                • 6.8.3 Skew fields over R
                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • 6.8.4 Tensor products
                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • 6.8.5 Splitting fields
                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • 6.8.6 Reduced norms and traces
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • 6.8.7 The Skolem-Noether Theorem
                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • 6.8.8 The Brauer group
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • Bibliography
                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • Index of Notations
                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • Index
                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • About the Author

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