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群、环、域导引（影印版）

样章
作者：
Fernando Q. Gouvêa
定价：
135.00元
ISBN：
978-7-04-056999-5
版面字数：
560.000千字
开本：
特殊
全书页数：
暂无
装帧形式：
精装
重点项目：
暂无
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
  • 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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