Algebra and Galois Theories

Introduction
Contents
About the Authors
1 Zorn's Lemma
1.1 Choice Functions
1.1.1
1.1.2
1.1.3 Commentary
1.1.4 Example 1: Existence of Non-measurable Sets
1.1.5 Example 2: The Banach-Tarski Paradox
1.1.6
1.2 Well-Ordered Sets
1.2.1 Ordered Sets
1.2.2 Upper and Lower Bounds
1.2.3
1.2.4 Examples
1.2.5
1.2.6
1.3 τ-Chains
1.3.1
1.3.2 Example
1.3.3
1.3.4
1.3.5
1.4 Inductive Sets and Zorn's Lemma
1.4.1
1.4.2 Examples
1.4.3
1.4.4
1.5 Applications of Zorn's Lemma
1.5.1
1.5.2
1.5.3
1.6 Noetherian Ordered Sets
1.6.1
1.6.2
1.6.3
1.7 Tychonoff's Theorem
1.7.1 Filters
1.7.2 Ultrafilters
1.7.3 Characterization of Compact Sets
1.7.4 Image Filter
1.7.5
1.7.6
1.7.7
Reference
2 Categories and Functors
2.1 Categories
2.1.1
2.1.2 Examples
2.1.3 Universe
2.1.4
2.1.5 Examples
2.1.6 Definition
2.1.7 Definition
2.2 Functors
2.2.1 Definition
2.2.2 Examples
2.2.3 Definition
2.2.4 Examples
2.2.5 Category of Categories
2.2.6 Another Example of a Functor
2.2.7
2.3 Morphisms of Functors
2.3.1 Definition
2.3.2 Examples
2.3.3 Functor Category
2.3.4 Definitions
2.3.5 Equivalence of Categories
2.3.6
2.3.7
2.4 Representable Functors
2.4.1 Notation
2.4.2 Definition
2.4.3
2.4.4 Definition
2.4.5 Comments. Universal Problems
2.4.6 Definition
2.4.7 Example of a Universal Problem
2.4.8 Other Examples of Representable Covariant Functors
2.4.9 Examples of Representable Contravariant Functors
2.4.10
2.4.11 Uniqueness of the Solution of the Universal Problem
2.4.12
2.5 Products and Inverse Limits
2.5.1 Definition
2.5.2 Examples
2.5.3
2.5.4 Definition
2.5.5 Definition
2.5.6 Definition
2.5.7 Definition
2.5.8 Definition
2.5.9 Example
2.5.10
2.5.11 Examples
2.5.12 Morphisms of Inverse Systems
2.5.13 Cofinal Sets
2.5.14 Inverse Limits of Compact Spaces
2.5.15
2.6 Sums and Direct Limits
2.6.1 Definition
2.6.2 Examples
2.6.3 Definition
2.6.4 Definition
2.6.5 Direct Limits of Sets
2.6.6
2.6.7
2.6.8 Direct Limit of Categories
2.7 Adjoint Functors
2.7.1 Definition
2.7.2
2.7.3
2.7.4 Examples
2.7.5
2.7.6 Topology and Algebra
2.7.7 Spectra and Gelfand Transforms
2.7.8
2.8 Profinite Spaces
2.8.1 Pro-objects of a Category
2.8.2 Totally Disconnected Spaces
2.8.3
2.8.4
2.8.5
2.8.6
2.9 Profinite Groups
2.9.1
2.9.2
2.9.3
2.9.4
2.9.5
2.9.6
2.9.7
2.9.8
Reference
3 Linear Algebra
3.1 Integral and Principal Ideal Domains, Reduced Rings
3.1.1 Integral Domains
3.1.2 Examples of Integral Domains
3.1.3 Field of Fractions
3.1.4 Prime Ideals and Maximal Ideals
3.1.5
3.1.6 Nilpotent Elements, Nilradical
3.1.7 Reduced Rings
3.1.8
3.1.9 Principal Ideal Domains
3.1.10 Euclidean Rings
3.1.11 Examples
3.2 Unique Factorization Domains
3.2.1 Monoids
3.2.2 Divisibility
3.2.3 Support of a Family
3.2.4 Free Commutative Monoids on a Set
3.2.5 Free Commutative Monoids
3.2.6
3.2.7 Monoids Associated to Integral Domains
3.2.8 Unique Factorization Domains
3.2.9
3.2.10
3.2.11
3.3 Modules
3.3.1
3.3.2
3.3.3 Submodules
3.3.4 Torsion
3.3.5 Generating Families
3.3.6 Quotient Modules
3.3.7 Canonical Factorization
3.3.8 Exact Sequences
3.3.9
3.3.10 Complement Submodules
3.3.11 Projections
3.3.12 Split Exact Sequences
3.3.13 Direct Products and Sums
3.3.14 Direct and Inverse Limits
3.3.15 Modules of Homomorphisms
3.3.16 Left Exactness of the Functor Hom
3.3.17 Multilinear Maps
3.3.18 Signature of a Permutation
3.3.19 Algebras
3.4 Free Modules, Matrices
3.4.1 Free Modules
3.4.2 Examples
3.4.3 Counterexamples
3.4.4 Alternating n-Linear Forms
3.4.5 Uniqueness of the Dimension
3.4.6 Free Module on a Set
3.4.7 Projectivity of Free Modules
3.4.8 Projective Modules
3.4.9 Matrices
3.4.10 Matrices Representing a Homomorphism with Respect to Given Bases
3.4.11 Determinants
3.4.12 Minors
3.4.13
3.4.14
3.4.15
3.4.16 Presentations
3.4.17
3.4.18
3.5 Modules over Principal Ideal Domains
3.5.1
3.5.2 Adapted Bases for Submodules
3.5.3 Content
3.5.4
3.5.5 Elements of Content 1
3.5.6 Elements of Minimal Content in a Submodule
3.5.7
3.5.8
3.5.9
3.5.10 Elementary Divisors
3.5.11 Primary Decomposition
3.6 Noetherian Rings
3.6.1
3.6.2 Examples
3.6.3 Counterexamples
3.6.4 Minimal Prime Ideals
3.6.5 Associated Prime Ideals
3.6.6
3.6.7
3.6.8 Remark
3.7 Polynomial Algebras
3.7.1 Monoid Algebra
3.7.2 Polynomial Algebras
3.7.3 Notation
3.7.4
3.7.5
3.7.6 Degree of Polynomials
3.7.7
3.7.8 Polynomials with Coefficients in a UFD
3.7.9
3.7.10 Substitution, Polynomial Functions
3.7.11 Resultants
3.7.12 Discriminants
3.7.13 Algebraic Sets
3.7.14 Principle of Extension of Identities
3.7.15 Examples
3.8 Tensor Products
3.8.1
3.8.2
3.8.3 Functoriality
3.8.4 Symmetry
3.8.5 Identity Object
3.8.6 Associativity
3.8.7 Right Exactness, Distributivity, Passing to Direct Limits
3.8.8 Tensor Product of Free Modules
3.8.9 Tensor Product of Matrices
3.8.10 Tensor Product of Cyclic Modules
3.8.11 Examples of Tensor Products over mathbbZ
3.8.12 Extension of Scalars
3.8.13 Extension of Scalars to the Field of Fractions
3.8.14 Rank of a Module
3.8.15 Tensor Products of Algebras
3.8.16
3.8.17
3.8.18 Examples
3.9 The Kronecker Homomorphism
3.9.1
3.9.2 Functoriality
3.9.3
3.9.4 Specific Cases When is an Isomorphism
3.9.5 Specific Cases When α is Bijective
3.9.6 The Noetherian Case
3.9.7 The Principal Ideal Domain Case
3.9.8 Image of α
3.9.9 Contraction and Composition
3.9.10 Trace
3.9.11
3.9.12 Trace and Extension of Scalars
3.9.13
3.9.14 Computation of the Rank of a Tensor
3.10 Chain Complexes
3.10.1 Graded Module
3.10.2 Chain Complexes
3.10.3 Morphisms of Chain Complexes
3.10.4 Homotopic Morphisms
3.10.5
3.10.6 Connecting Homomorphism
3.10.7 Diagramme du serpent
3.10.8 Mapping Cylinder
3.10.9 Resolutions
3.10.10 Projective Resolutions
3.10.11 Injective Modules
3.10.12
3.10.13
3.10.14 Injective Resolutions
References
4 Coverings
4.1 Spaces over B
4.1.1
4.1.2 Change of Basis
4.1.3 Hausdorff Spaces over B
4.1.4 Etale Spaces over B
4.1.5 Proper Spaces
4.1.6
4.1.7 Fibres
4.1.8 Example. The Möbius Strip
4.1.9 Fibre Bundle over a Segment
4.1.10 Partition Subordinate to a Cover
4.1.11 Gluing Trivializations
4.2 Locally Connected Spaces
4.2.1
4.2.2 Examples
4.2.3 Counterexamples
4.2.4
4.2.5
4.2.6
4.2.7 Embeddings in a Banach Space
4.2.8
4.3 Coverings
4.3.1
4.3.2
4.3.3
4.3.4 Examples
4.3.5
4.3.6
4.3.7
4.3.8 Subcoverings, Quotient Coverings
4.3.9 Examples
4.3.10
4.3.11
4.3.12
4.3.13
4.3.14
4.3.15
4.3.16
4.3.17
4.3.18 Coverings of Products
4.3.19 Product of Simply Connected Spaces
4.4 Universal Coverings
4.4.1
4.4.2
4.4.3
4.4.4
4.4.5
4.4.6 Product Covering
4.4.7
4.5 Galois Coverings
4.5.1
4.5.2 Examples
4.5.3 The Functor S
4.5.4
4.5.5
4.5.6
4.5.7
4.6 Fundamental Groups
4.6.1
4.6.2
4.6.3 Comparison of π1(B,b0) and AutB(E)
4.6.4 Functoriality
4.6.5
4.6.6 Basepoint-Change
4.6.7
4.6.8
4.6.9 Dictionary
4.6.10
4.6.11
4.6.12 Fundamental Group of a Product
4.7 Van Kampen's Theorem
4.7.1 Categorical Preliminaries
4.7.2
4.7.3
4.7.4 Free Groups on Sets
4.7.5
4.7.6 Amalgamated Sums
4.7.7
4.8 Graphs. Subgroups of Free Groups
4.8.1 Graphs
4.8.2
4.8.3
4.8.4
4.8.5
4.8.6
4.8.7
4.8.8
4.8.9
4.9 Loops
4.9.1 Paths
4.9.2 Standard Paths
4.9.3 Juxtaposition
4.9.4 Lifting of Paths
4.9.5
4.9.6 Poincaré Group
4.9.7 Functoriality
4.9.8 Poincaré Group of a Product
4.9.9 The Poincaré Group and the Fundamental Group
4.9.10 Surjectivity
4.9.11
4.9.12 Construction of Universal Coverings as Spaces of Paths
4.9.13
4.9.14 Proof of Theorem 4.9.9
4.9.15
Reference
5 Galois Theory
5.1 Extensions
5.1.1 Finite Algebras
5.1.2
5.1.3
5.1.4
5.1.5
5.1.6
5.1.7
5.1.8 Transcendental Elements
5.2 Algebraic Extensions
5.2.1
5.2.2
5.2.3
5.2.4 Example. d'Alembert's Theorem
5.2.5
5.2.6
5.2.7
5.2.8 Characteristic of a Field and the Frobenius Endomorphism
5.2.9 Finite Fields
5.2.10
5.3 Diagonal Algebras
5.3.1 Algebra of Functions on a Finite Set
5.3.2 The Gelfand Transform
5.3.3
5.3.4 Example. (The Fourier Transform for mathbbZ/(n).)
5.3.5
5.3.6
5.4 Etale Algebras
5.4.1
5.4.2
5.4.3
5.4.4
5.4.5
5.4.6
5.4.7
5.4.8
5.4.9
5.4.10 Example
5.4.11 Primitive Element Theorem
5.5 Purely Inseparable Extensions
5.5.1
5.5.2
5.5.3
5.5.4
5.5.5
5.5.6 Characteristic 0 Conventions
5.6 Finite Galois Extensions
5.6.1
5.6.2
5.6.3
5.6.4
5.6.5 Decomposition Field
5.6.6
5.6.7
5.7 Finite Galois Theory
5.7.1
5.7.2
5.7.3
5.7.4
5.7.5
5.8 Solvability
5.8.1
5.8.2
5.8.3
5.8.4 Solvable Extensions
5.8.5 Separable Solvable Extensions
5.8.6 Solvable Groups
5.8.7 The Groups Sn and An
5.8.8 Simplicity of An for n5
5.8.9
5.8.10
5.8.11 Example of a Non Solvable Extension of mathbbQ
5.9 Infinite Galois Theory
5.9.1
5.9.2 Profinite Structure of G
5.9.3 The Functor S
5.9.4
5.9.5 Dictionary
5.9.6
Reference
6 Riemann Surfaces
6.1 Riemann Surfaces, Ramified Coverings
6.1.1 Riemann Surfaces
6.1.2 Example. The Riemann Sphere
6.1.3 Analytic Maps
6.1.4
6.1.5
6.1.6
6.1.7 Ramified Coverings
6.1.8
6.1.9
6.1.10
6.1.11
6.1.12
6.2 Ramified Coverings and Etale Algebras
6.2.1 Separation Theorem
6.2.2 Analytic Criterion for Meromorphism
6.2.3 The Functor mathcalM
6.2.4
6.2.5 Construction of a Covering mathcalS(E, ζ)
6.2.6 Functoriality
6.2.7
6.2.8
6.2.9 Proof of Theorem 6.2.4
6.2.10 Dictionary
6.2.11
6.2.12
6.3 Extensions of mathbbC with Transcendence Degree 1
6.3.1
6.3.2 Meromorphic Functions on the Riemann Sphere
6.3.3 Example: Homographies
6.3.4 Transcendence degree of mathcalM(X)
6.3.5
6.4 Determination of Some Galois Groups
6.4.1 Free Profinite Groups
6.4.2 The Homomorphism θx
6.4.3 Galois Group of ΩΔ
6.4.4 Fundamental Groups with Respect to Germs of Paths
6.4.5 Galois Group of the Algebraic Closure of mathcalM(B)
6.4.6 Galois Group of the Algebraic Closure of mathbbC(Z)
6.5 Triangulation of Riemann Surfaces
6.5.1 Definition of a Triangulation
6.5.2 Direct C1 Triangulations
6.5.3 Existence of Triangulations of the Riemann Sphere
6.5.4 Lifting of a Triangulation
6.5.5
6.5.6
6.6 Simplicial Homology
6.6.1 Chain Complex Associated to a Triangulation
6.6.2 Connected Case
6.6.3 Barycentric Subdivision
6.6.4
6.6.5 Dual of a Chain Complex
6.6.6
6.6.7 Remark
6.6.8
6.6.9 Intersection Product
6.6.10
6.6.11
6.6.12 The Euler-Poincaré Characteristic
6.6.13 The Riemann–Hurwitz Formula
6.6.14 Genus, Uniformization
6.7 Finite Automorphism Groups of Riemann Surfaces
6.7.1 Quotient Riemann Surface
6.7.2 Genus g ge2 Case
6.7.3 Genus 1 Case
6.7.4 Genus 0 Case
6.8 Automorphism Groups: The Genus 2 Case
6.8.1 Homogeneity of Topological Surfaces
6.8.2
6.8.3 Coalescence of Ramifications
6.8.4
6.8.5 Coverings Ramified at a Single Point
6.8.6
6.8.7
6.9 Poincaré Geometry
6.9.1 Homographic Transformations of Circles
6.9.2
6.9.3 Automorphisms of mathbbD
6.9.4 Riemannian Metrics
6.9.5 Poincaré Metric
6.9.6
6.9.7 Poincaré Circles
6.9.8 Geodesic Paths
6.9.9 Geodesics
6.9.10
6.9.11 Isometries of mathbbD
6.9.12
6.9.13
6.9.14 Geodesically Convex Sets
6.9.15 Geodesic Triangles
6.9.16 Sum of the Angles of a Geodesic Triangle
6.9.17 Finiteness of the Automorphism Group (gge2)
6.10 Tiling of the Disk
6.10.1 The Dihedral Group
6.10.2 A Tiling of mathbbD
6.10.3 Construction of E
6.10.4 The Star of a in E
6.10.5
6.10.6
6.10.7
6.10.8
6.10.9 Proof of Theorem 6.10.2
6.10.10 Crossed Products
6.10.11
6.10.12
6.10.13
6.10.14
6.10.15
6.10.16
6.10.17
6.10.18
References
7 Dessins d'Enfants
7.1 Definability
7.1.1 Defining Polynomials
7.1.2 Definability of a Ramified Covering
7.1.3 Semi-definability
7.1.4 Definability for Riemann Surfaces
7.2 Belyi's Theorem
7.2.1 Statement of the Theorem
7.2.2 Implication (i) (ii)
7.2.3 Density
7.2.4 Polynomials with k Distinct Roots
7.2.5 Implication (ii) (i)
7.2.6 The Action f!
7.2.7 Rational Critical Values
7.2.8 Reduction of the Number of Critical Values
7.3 Equivalence Between Various Categories
7.3.1 Ramification Locus of an Algebra
7.3.2 Transfer of Δ
7.3.3
7.3.4 The Category mathscrA
7.3.5 Characterization of the Elements of an Etale Algebra
7.3.6 The Categories mathscrE and mathscrV
7.3.7 A Diagram of Categories
7.3.8 Cyclic Order
7.3.9 Graphs
7.3.10 The Category mathscrH
7.3.11 The Functor θ:R-2.5muR1pt. tomathscrH
7.3.12 The Functor η
7.3.13 Action of mathbbG on mathscrH
7.3.14 A Numerical Equivalence Class
7.3.15 The Extended Valency Function
7.4 Belyi Polynomials
7.4.1 Ramified Polynomial Coverings
7.4.2
7.4.3 Belyi Polynomials
7.4.4 Statement of the Faithfulness Theorem
7.4.5 Comparing Factorizations
7.4.6 Proof of the Faithfulness Theorem
7.5 Two Examples
7.5.1 First Example: Setting of the Problem
7.5.2 The Polynomial Qc
7.5.3 Study of Q = Qc
7.5.4 Interpretation of These Results
7.5.5 Second Example: ``Leila's bouquet''
7.5.6 Computing Discriminants
7.5.7 The Invariant δa-3.5mul-3.7mug
7.5.8 Definition of δt-3.5muo-3.7mup(P)
7.5.9 Notation
7.5.10 A Covering Property
7.5.11 The Tangent Linear Map ρ
7.5.12 A Properness Property
7.5.13 A Uniqueness Property
7.5.14 Topological Numbering
7.5.15 The Invariant θt-3.5muo-3.7mup(Q)
7.5.16 Relation Between δt-3.5muo-3.7mup(P) and θt-3.5muo-3.7mup(Q)
References
Appendix Index of Notation
Appendix Bibliography
Notation Defined in Text
Index