Abstract Algebra and Discrete Mathematics

By Karl Dahlke, Copyright © 2017

This e-book is an introduction to combinatorics, number theory, topology, groups, rings, fields, modules, algebraic and integral extensions, noncommutative algebra, algebraic number theory, algebraic geometry, algebraic topology, and even more. The focus is on breadth rather than depth. Excellent books already exist for any one of these topics in detail, and I don't want to reinvent that wheel. Instead, this book knits them all together, providing a foundation for each topic in turn. By analogy, you might point your backyard telescope to every corner of the galaxy, in an effort to comprehend its scope, beauty, and diversity. You might not understand the Crab Nebula in all its detail, but you should walk away with an appreciation for the vastness and the wonder of the galaxy, in this case, the galaxy of modern mathematics. If, from time to time, you find yourself saying, "How did anybody ever think of that?", then I have succeeded.

Each chapter builds on the information that has gone before, and forward references are rare, though they do happen from time to time. I hope this on-line book is more accessible than a sea of disconnected web pages, which is the hallmark of most math websites. Send along your questions or feedback.

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Contents

0. Prolog
1. Data Structures
2. Set Theory
3. Combinatorics
4. Number Theory
5. Euclidean Domains
6. Difference Equations
7. Fields
8. Finite Fields
9. Cyclotomic Extensions
10. Groups
11. Groups Acting on Sets
12. Similar Matrices and Jordan Canonical Form
13. Finite Simple Groups
14. Generating Functions
15. Galois Groups and Extensions
16. Straightedge Compass Construction
17. Solvable Groups, Nilpotent Groups, and the Composition Series
18. Solvable Extensions and Polynomials
19. Separable Extensions
20. Category Theory
21. Free Groups
22. Permutation Groups
23. Rings and Ideals
24. Principal Ideal Domains
25. Modules
26. Noetherian and Artinian Modules and Rings
27. Modules over a PID
28. Krull Schmidt
29. Fractions
30. Radical Ideals in a Commutative Ring
31. Primary Ideals and Laskerian Rings
32. Simple and Semisimple Rings and Modules
33. The Jacobson Radical
34. Radical Ideals in a Noncommutative Ring
35. Local Rings
36. Twisted Rings
37. Division Rings
38. Quadratic Forms
39. Orbital Mechanics
40. Matrix Polynomials
41. Point Set Topology
42. Metric Spaces
43. Compact Sets
44. Banach and Hilbert Spaces
45. Topological Groups and Modules
46. Topological Dimension
47. Spec R
48. Graded Rings
49. Projective, Injective, Tensor Product
50. R Algebras
51. Integral Extensions
52. Valuation Rings
53. Dedekind Domains
54. Elliptic Curves
55. Lattice in n Space
56. Integral Rings
57. Algebraic Number Theory
58. Quadratic Number Fields
59. Continued Fractions
60. Cyclotomic Number Fields
61. Generalized Reciprocity
62. Local Fields

What if you are interested in pythagorean triples, but you're not sure which chapter to look in? Number Theory is a good guess, but in fact those triples cannot be analyzed without the machinery of Euclidean Domains. Nobody wants to dip into each chapter, searching for a particular topic. The following master index presents every section of every chapter in order. Scroll through to get a feel for each chapter and what it contains, or use the control-F search function in your browser to look for particular keywords on this page, then jump straight to that section. Realize however that you may need to start reading at the top of the chapter for context and clarity.

Data Structures

• Properties of Arithmetic
• Modular Math
• Homomorphism
• Casting out Nines
• Credit Card Numbers and the Luhn Algorithm
• Monomorphism, Epimorphism, Isomorphism
• Discrete Logs
• Binary Search
• Function Composition
• Permutations
• Even and Odd Permutations
• Matrices
• Linear Functions
• Matrix as Function
• Determinant
• Gaussian Elimination
• Elementary Row Operations
• The Determinant of the Product
• Matrix Identities and Inverses
• Orthogonal
• Determinant equals Volume
• The Gram Schmidt Process
• The Shoelace Formula
• Orthonormal and Rotations
• Cross Product
• Vandermonde Matrix
• A Matrix of Matrices
• Complex Numbers and the Gaussian Integers
• Eisenstein Integers
• Quaternions
• Half Integer Quaternions
• Projective Space and an Interesting Homomorphism
• The Square Root of a Rotation
• The Hairy Ball Theorem
• Generalized Euclidean Space
• Polynomials
• Synthetic Division, Roots, and GCD
• Formal Derivative
• Power and Laurent Series
• p-adic Numbers

Set Theory

• Sets and Axioms
• Direct Sum and Product
• Relations and Functions
• The Pigeonhole Principle
• Ordinals and Cardinals
• Countable, Uncountable, and Diagonalization
• The Schroder Bernstein Theorem
• Cardinal Math
• Transfinite Induction
• Hydrogen
• The Cantor Set
• Space Filling Curve

Combinatorics

• Independent Events
• Permutations
• Circular Permutations
• Combinations
• Pascal's Triangle
• Binomial / Multinomial Theorem
• Binomial Theorem and Differentiation
• Continuous Binomial Theorem
• Factors of p in n!
• Poker Hands
• The Birthday Paradox
• Sampling with or without Replacement
• The Monty Hall Dilemma
• Inclusion Exclusion
• No-Match Permutations
• The Marriage Problem
• Catalan Numbers
• Arithmetic Mean, Geometric Mean

Number Theory

• Euclid's GCD Algorithm and Unique Factorization (UFD)
• Infinitely Many Primes
• Irrational Roots
• Linear Combination Yields the GCD
• A Lattice of Solutions
• Modular Inverse
• The Chinese Remainder Theorem, Integers
• φ and σ Functions
• Mersenne and Fermat Primes
• Möbius Function
• ζ(2)
• ζ(4), ζ(6), etc
• The Odds of being Coprime
• Fermat's Little Theorem
• RSA Encryption
• Wilson's Theorem
• Primitive Root
• The Units mod m
• The Pseudoprime Test and Carmichael Numbers
• A Proof of Primality
• Quadratic Residues and the Legendre Symbol
• Quadratic Reciprocity
• The Jacobi Symbol
• Pollard Rho Factorization

Euclidean Domains

• Euclidean Domain
• Linear Combination Yields the GCD
• Gaussian Integers
• Gaussian Primes
• Gaussian Integers mod p
• Pythagorean Triples
• Sum of Squares
• Two Squares and a Fourth Power
• Adjoining the Square Root of -2
• Adjoining the Square Root of -5
• Eisenstein Integers
• Adjoining the Square Root of -3
• Fermat's Last Theorem
• Fermat's Last Theorem, n = 3
• Fermat's Last Theorem, n = 4
• Mordell and Euler Conjectures
• Adjoining the Eighth Roots of 1
• Quaternion GCD
• Polynomials over a Field form a Euclidean Domain
• Partial Fractions
• Gauss' Lemma
• Eisenstein's Criterion
• Fractions in Lowest Terms

Difference Equations

• Difference sequence / Difference Equation
• Existence and Uniqueness
• Dimensionality
• Polynomial Solutions
• Sum of n^th Powers
• Homogeneous Linear Equation with Constant Coefficients
• Phase Shifted Functions
• The Golden Ratio
• The Fibonacci Sequence
• Fibonacci Commutes with GCD
• The Fibonacci Matrix

Fields

• Fields
• Frobenius Homomorphism
• Ordered Field
• Division Ring
• Field Homomorphism Embeds
• Basis and Dimension
• Dimension of Kernel and Image
• The Dimension of a Tower of Extensions
• Field Extension
• Adjoined Element
• Finitely Generated
• Splitting Field
• Closed Field
• Extending the Isomorphism
• Indistinguishable Roots
• Normal Field Extension
• The Composition of Normal Extensions
• Uniform Factoring
• Transcendent Space

Finite Fields

• Finite Fields
• Primitive Root
• -1 and 2 as Squares
• Unique Field for each Prime Power
• The Automorphisms and Subfields of F
• Circular Shift of the Roots
• Finding an Irreducible Polynomial
• Irreducibles by Degree
• Irreducibles by Squares and Shifted Squares
• Polynomial Factorization
• The Norm Distribution Theorem
• Counting Matrices
• The Structure of O₂
• O Acts on Unit Vectors
• Quaternions and Matrices are Isomorphic
• Algebraic Closure mod p

Cyclotomic Extensions

• Angle Addition Formula
• Double Angle Formula
• Half Angle Formula
• Demoivre's Formula
• Cyclotomic Extensions
• Primitive n^th Root
• Automorphisms
• Splitting xⁿ-1
• ζ Polynomials
• ζ Irreducible over Z
• ζ Irreducible over Z[i]
• Conjugates and Norm
• Ratio Units
• 1-y, lying over n

Groups

• Groups
• Cancellation and Order
• Subgroups and Cosets
• Kernel and Quotient
• The Group of Automorphisms of G
• Inner Automorphisms
• Direct Product, Direct Sum
• Semidirect Product
• Wreath Product
• Intersection and Join
• Center, Centralizer, Normalizer
• The Correspondence / Isomorphism Theorems
• Dihedral Group
• General Linear, Special Linear
• Coliniation
• Symmetric, Alternating

Groups Acting on Sets

• Group Action
• Orbit, Stabilizer
• Translation, Conjugation
• Macay's Theorem
• The Fixed Point Principle
• Transitive, Doubly Transitive
• The Equal Orbits Principle
• The Strong Cayley Theorem
• p Groups and p Sylow Subgroups
• The Congruent Index Principle
• The First Sylow Theorem
• The Second Sylow Theorem
• The Third Sylow Theorem
• Sylow Intersection
• The Normalizer of the Normalizer
• The Structure of a Finite Abelian Group
• Cyclic Multiplicative Group
• Groups of Order p² and p³
• Quaternion Groups of Order 8 and 24
• Metacyclic Groups
• The Burnside Counting Theorem
• The Burnside Polya Theorem
• Colored Necklaces
• Regular / Platonic Solids
• Euler's Formula
• The Dual of a Polyhedron
• Semiregular / Archimedean Solids
• Rotations and Reflections of the Semiregular Solids
• Descartes' Angular Defect Theorem
• Quasiregular Solids

Similar Matrices and Jordan Canonical Form

• Conjugacy Class of Similar Matrices
• Eigen Vector, Eigen Value
• Similarity and Eigen Vectors
• Diagonalizable
• Schur's Theorem
• Trace and Norm
• Block Diagonal
• Direct Sum, Invariant, Complementary
• Nilpotent Transformation
• Jordan Canonical Form
• Complementary Spaces in a Singular Transformation
• A Jordan Form for Each Conjugacy Class

Finite Simple Groups

• Overview
• Alternating Groups
• A Simple Group Contains Even Permutations
• Generated by p Cycles
• Simple Groups up to 200
• p Cycles and Normality
• Group of Order 168
• Special Linear, Center
• p Sylow Subgroups
• Eigen Values of 1, All or None
• Eigen Values in F
• Special Linear Group is Simple, n = 2
• Special Linear Group is Simple

Generating Functions

• Generating Function
• Context Free Grammar
• Triangulating Polygons
• Approximating cat(n)
• Polya's Enumeration Theorem
• p(n)
• Counting Cycle Decompositions
• The Power of Exponentiation
• Rooted Trees

Galois Groups and Extensions

• History
• Galois Group
• Subgroup Subfield Correspondence
• Stable Subfield
• Galois is Normal and Separable
• A Quadratic Extension is Galois
• Galois Subfield has Quotient Galois Group
• The Composition of Galois Extensions
• Every Finite Group is Galois
• A Transcendental Galois Extension
• Fractional Linear Transforms

Straightedge Compass Construction

• Background
• Rational Coordinates
• A Field of Distances
• A Tower of Quadratic Extensions
• Angle Trisection
• The Regular n-gon

Solvable Groups

• Subnormal Series, Solvable Group
• Refinement
• Composition Series
• Subgroup of a Solvable Group
• Commutator subgroup, Abelianization
• Commutator subgroups and Solvability
• Central Series, Nilpotent
• The Center and a Normal Subgroup Intersect
• Finite Nilpotent Group
• The Intersection of Maximal Subgroups is Nilpotent
• Zassenhaus
• Jordan Holder
• Uniserial

Solvable Extensions and Polynomials

• Connections
• Etimology
• Quadratic Formula
• Cubic Formula
• Quartic Formula
• The Galois Group of p(x)
• Cyclic Extension, Solvable Extension
• Field Automorphisms are Linearly Independent
• Hilbert's Theorem 90
• Irreducible or Split
• p Cyclic, Characteristic p
• p Cyclic, Characteristic not p
• Solvable Polynomials and Groups
• An Unsolvable Quintic
• Finite Fields

Separable Extensions

• Definitions
• Characteristic 0
• Finite Fields
• Separable and Galois
• Separable Subfield
• The Primitive Element Theorem
• Purely Inseparable
• Purely Inseparable Subfield
• Infinitely Many Subfields Inside
• The Fundamental Theorem of Algebra
• Base Change / Compositum
• Linearly Disjoint
• Direct Product of Galois Groups
• Inseparable Degree
• Trace and Norm
• Norm and Irreducibility

Category Theory

• Category, Object, Morphism
• Free Object
• Monic, Epic
• Injective, Projective
• Initial, Terminal
• Commutative Diagram
• Product, Coproduct
• Limit, Colimit
• Pushout, Pullback
• Direct Limit, Inverse Limit
• Graded Category
• Functor

Free Groups

• A Free Group
• Counting Words
• Relators, Relations, Presentation
• Equivalent Presentations
• Product and Coproduct
• Free Abelian
• Larger Rank Inside a Smaller Rank
• Matrix Representations
• Schreier Nielsen Rewriting
• The Subgroup of a Free Group is Free
• Locally Finite and Torsion

Permutation Groups

• Permutation Puzzles
• Parity
• Strong Generators
• Path to Start
• Finding Strong Generators
• Orbits and Normal Subgroups
• Other Subgroups

Rings and Ideals

• Definition
• Units and Associates
• Integral Domain
• Subring
• Ideal
• Ring Homomorphism
• Ideals in a Matrix Ring
• Quaternion Primes over Primes
• Every Integer is the Sum of 4 Squares
• The Chinese Remainder Theorem, Rings
• Irreducible and Prime Elements
• Maximal and Principal Ideals
• Prime Ideals
• Direct Sum, Direct Product
• Prime Ideal Correspondence
• Intersection and Prime Ideals
• The Union of Prime Ideals
• Semiprime Ideals
• Prime and Semiprime Rings
• Prime Ideal Outside of a Multiplicative Set
• Nilpotent, Idempotent
• Maximal Ideals and Fields
• Extension and Contraction
• Maximal Infinitely Generated Ideal

Principal Ideal Domains

• Euclidean Domain → PID → UFD
• Principal Prime Ideals
• 5 Criteria for a UFD
• Quotient of a UFD
• Bezout's Identity
• A PID that is not a Euclidean Domain
• Minimal Nonzero Prime Ideal
• Arithmetic Ring

Modules

• Definition
• Unitary Module
• Submodule
• Module Homomorphism
• Ring of Endomorphisms
• Bimodule
• Free Module, Basis
• The Rank of a Free Module
• Finitely Generated, Finitely Presented
• A Product of Rings and Modules
• Conductor Ideal, Annihilator
• Simple Module
• Jordan Holder

Noetherian and Artinian Modules and Rings

• Noetherian, Artinian
• Chains of Sets
• Finitely Generated Submodules
• Finitely Generated Prime Ideals
• Kernel, Quotient
• Two Quotients
• Finitely Generated over a Noetherian Ring
• Jordan Holder
• Endomorphisms
• An Artinian Domain is a Division Ring
• Hilbert's Basis Theorem
• Finitely Generated Ring
• Formal Power Series
• Factorization Domain
• Finitely Many Minimal Prime Ideals
• An Asymmetric Ring
• Prime in a Principal Ideal

Modules over a PID

• Overview
• The Submodule of a Free Module is Free
• Torsion, Order
• Finitely Generated and Torsion Free is Free
• The Structure of a Finitely Generated Module

Krull Schmidt

• Overview
• ACC, DCC
• Decomposable
• Endomorphisms
• Fittings Decomposition Theorem
• Nilpotent
• Ring of Endomorphisms
• Unique Decomposition
• Two Representations

Fractions

• S inverse of R
• S inverse of M
• Ideal Correspondence
• Prime Ideal Correspondence
• Localization, Local Ring
• Saturated Sets and Prime Ideals
• The Saturation of an Ideal
• Mapping One Fraction Ring into Another
• The Structure of R/S/T
• The Non Zero Divisors
• Noetherian Fraction Ring
• The Intersection of Localizations
• Zero is a Local Property
• Induced Homomorphism
• Fractions of the Quotient Ring
• Isomorphic Quotients

Radical Ideals in a Commutative Ring

• Radical Ideal
• One Prime Ideal
• Intersection and Product
• Unit and Nilpotent Elements
• Unit and Nilpotent Polynomials
• Zero Divisor Polynomials
• Jacobson Radical
• Radicals in Polynomial Rings
• Reduced is a Local Property

Primary Ideals and Laskerian Rings

• Primary Ideal
• The Radical of a Primary Ideal
• The Finite Intersection of P Primary Ideals
• Primary Decomposition
• Associate Primes
• First Uniqueness
• Self Radical
• Minimal Primes over J
• Polynomial Ring
• pan(R)
• pan(R) in the Fraction Ring
• Saturation
• The Intersection of the Saturations of 0
• Minimal Primes and the Radical of the Saturation
• The Smallest P Primary Ideal
• Finitely Many Saturations
• Second Uniqueness
• An Embedded Prime
• Saturation Through a Single Element
• Finite Intersection Equals Finite Product
• Laskerian is l₁
• l₁ and Noetherian Implies Laskerian
• Laskerian is l₂
• l₁ and l₂ Implies Laskerian
• Noetherian Implies Laskerian
• Primary Modules

Simple and Semisimple Rings and Modules

• The Endomorphisms of a Simple Module
• Semisimple
• Characterizing a Semisimple Module
• Semisimple Inheritance
• Noetherian and Artinian
• The Artin Wedderburn Theorem
• The Center of a Semisimple Ring
• A Simple Ring that is Not Artinian

The Jacobson Radical

• Definition and Equivalent Characterizations
• Nil Ideals in jac(R)
• Jacobson Semisimple
• Left Artinian implies J is Nilpotent
• Criteria for a nil Jacobson Radical
• Jacobson Semisimple, Left Semisimple, and Left DCC
• Semiprimary
• Artinian Implies Noetherian
• Nakiama's Lemma
• Maximal Ideals in an Artinian Ring
• Semilocal Ring
• The Jacobson Radical of a Subring
• Jacobson Radical of the Matrix Ring
• Von Neumann Ring
• Boolean Ring

Radical Ideals in a Noncommutative Ring

• Nil Ideal
• A Prime Reduced Ring
• m System, n System
• Radical Ideal
• Lower Nil Radical
• The Lower Nil Radical of a Matrix Ring
• The Lower Nil Radical of a Polynomial Ring
• Brauer's Lemma
• Semisimple = Semiprime with DCC
• One Sided Nil Radical
• Kothe's Conjecture
• Left Nilpotent Ideal
• String Representable
• Noetherian implies Kotherian
• Noetherian and Nil Radicals
• Brown Macoy Radical

Local Rings

• Local Ring
• Dedekind Finite
• Examples
• Strongly Indecomposable
• ACC and DCC
• Krull Schmidt
• A Finite Product of Maximal Ideals
• Noetherian and Dimension 0
• Semilocal
• The Structure of an Artinian Ring
• The Structure of an Artinian Local Ring

Twisted Rings

• Twisted Ring
• Zero Divisors, Nilpotents, Units
• Noetherian
• The Roots of a Polynomial
• Ideals and Generators
• Dedekind Domain
• The Chinese Remainder Theorem, Twisted
• Generalized Quaternions

Division Rings

• Division Ring
• A Finite Division Ring is a Field
• Finite Subgroups are Cyclic
• Algebraic over a Field
• Algebraic over the Reals
• Herstein's Lemma
• Commuting with the Commutators
• Normal is Central
• Central Multiplicative Commutators
• Generated by the Commutators
• Center of the Multiplicative Group
• Herstein's Little Theorem
• Infinite Centralizer
• Centrally Finite / Infinite

Quadratic Forms

• Introduction
• Conic Sections
• Ellipse
• Hyperbola
• Parabola
• Discriminant
• So - Why are They Called Conic Sections Anyways
• Ice Cream Cone Proof
• Parabolic Mirror
• Elliptical Mirror
• A Hall of Mirrors
• Hyperbolic Mirror
• Surfaces in 3 Space
• Translation
• Rotation
• Normal Matrix
• Hermitian Operator
• Intersecting with a Plane
• Unchanging Sign
• Extremal Values
• Which Direction is the Fastest

Orbital Mechanics

• Ptolemy
• Kepler's First Law
• Conservation of Momentum
• One Body Problem
• Two Body Problem
• Kepler's Second Law
• A Path in Polar Coordinates
• Kepler's Second Law (Proof)
• Kepler's First Law (Proof)
• Function of Time
• Kepler's Third Law
• Hooke's Law

Matrix Polynomials

• Introduction
• The Order of a Matrix
• Minimum Polynomial
• Cayley Hamilton
• Mathematics mod m(T)
• A Matrix of Functions
• Spectral Radius
• Analytic Matrix Function
• Exponential Matrix Function
• A System of Linear Differential Equations
• The Fundamental Theorem of Ordinary Differential Equations
• Homogeneous Linear Equation with Constant Coefficients

Point Set Topology

• In the Plane
• Definitions
• Interior, Closure, Boundary
• Subspace
• Dense, Separable
• A Base for the Topology
• First and Second Countable
• Order Topology, Linear Topology
• Separation Axioms
• Urysohn's Lemma
• Tietze's Extension Theorem
• The Limit of a Sequence
• Cluster Point
• Continuous Functions
• Locally Finite
• Pasting Continuous Functions Together
• Homeomorphism
• Product Space
• Coproduct Space
• Direct Sum
• Connected
• Path Connected
• Closure is Connected
• Locally Connected and Locally Path Connected
• Connected plus Locally Path Connected Implies Path Connected
• Connected Product
• Simply Connected
• Sewing Spaces Together
• Sewing a Space to Itself
• Mobius Strip
• Klein Bottle
• Quotient Space

Metric Spaces

• Like Real Space
• Distance Metric
• The Open Ball Topology
• Compressing the Metric
• Alternate Bases
• Product Space
• Triangular Inequality
• Cauchy Schwarz Inequality
• Generalized Euclidean Space
• Bounded, Diameter
• Continuous Functions
• Uniform Continuity
• Notation
• Urysohn's Metrization Criteria
• Cauchy Sequence
• Real Numbers
• Reals Form a Complete Metric Space
• The Completion of a Metric Space
• Least Upper Bound
• Intermediate Value Theorem
• Extending a Uniform Function
• The Lipschitz Constant
• The Lipschitz Constant and Derivatives
• Contraction Maps and the Attractor
• Nested Closed Sets

Compact Sets

• Open Cover
• Hausdorff Spaces
• Countably Compact
• Sequentially Compact
• Semicontinuous
• Compact Product Topology
• Compact in a Metric Space
• Closed and Bounded in Real Space
• Continuous Becomes Uniform
• Distance Between Functions
• Shrinking Diameters
• Separating Closed Sets
• The Tikhonov Product Theorem
• A Product of Sequentially Compact Sets
• Locally Compact
• Compactification
• Proper
• The Baire Category Theorem
• Nowhere Dense
• First and Second Category
• Uniform Boundedness Principle

Banach and Hilbert Spaces

• Normed Vector Space, Banach Space
• Continuous Module
• Equivalent Norms
• Quotient Space
• Bounded Linear Operator
• Bounded Operators form a Vector Space
• The Hahn Banach Theorem
• Continuous Map Becomes Bicontinuous
• Closed Graph Theorem
• Topological Vector Space
• Finite Dimensional is Euclidean
• Locally Compact Topological Vector Space
• Hilbert Space
• Dot Product is Continuous
• Orthonormal System, Hyperbasis
• Every Linear Continuous Function is Actually a Dot Product
• Inseparable Hilbert Space

Topological Groups and Modules

• Continuous Operators
• The Fundamental Group is Abelian
• Covering Space is a Continuous Group
• Filter Group
• The Intersection of the Local Base
• The Closure of a Subgroup is a Subgroup
• Completing a Continuous Group
• Different Filtrations, Same Completion

Topological Dimension

• Irreducible Set
• Chains and Components
• Noetherian
• Decomposition into Irreducible Components
• Dimension
• Correspondence Under Closure
• Generic Space
• Zariski Space

Spec R

• Zariski Topology
• Noetherian
• Lownil(R)
• Spec R is Compact
• Irreducible Sets and Dimension
• Noetherian Spectrum is Zariski
• Direct Product
• Examples
• A Contravariant Functor
• The Dimension of R[x]
• Going Up

Graded Rings

• Graded Ring / Module
• Noetherian
• Homogeneous Ideal
• Annihilators and Prime Ideals
• Proj R
• Localization
• A Contravariant Functor
• Beyond d₀

Projective, Injective, Tensor Product

• Projective, Injective
• Projective, Direct Sum, Direct Product
• Short Exact, Split Exact
• Summand of a Free Module
• The Dual of a Module
• Tensor Product
• Tensor Product is Commutative and Associative
• Tensor with a Direct Sum
• Tensor with an Ideal
• Tensor with a Quotient Ring
• Nonzero Tensor Product
• The Tensor of Two Functions
• Tensor and Short Exact
• Base Change
• Zero Tensor over a Local Ring
• Rank is Well Defined
• Generators that Span
• The Rank of a Free Submodule Inside Another Free Module
• A Free Submodule of Lesser Rank
• Flat Module
• Faithful Module
• Finite Flat Module
• Finite Flat over a Local Ring is Free
• Tensor with a Fraction Ring
• Fraction Ring is Flat
• Isomorphism is a Local Property
• Containment is a Local Property
• Tensor Product and Dual
• The Rank of a Finite Flat Module
• The Support of a Module
• Flat is a Local Property Over an Integral Domain
• The Invariant Factor Theorem

R Algebras

• R Algebra
• A Category of Algebras
• Tensor Product of Finitely Many Algebras
• Characterizing the Tensor Product of 2 Algebras
• Compositum

Integral Extensions

• Integral Extension
• Units in an Integral Extension
• Multiple of an Algebraic Element
• Finitely Generated = Integral
• Integrally Closed
• Multiplicatively Closed and Integrally Closed
• Localization and Integrally Closed
• Integral Quotient Ring
• Integral Fraction Ring
• Prime over Prime
• Integral and Jacobson Radical
• A Nonintegral Localization
• Integrally Closed is a Local Property
• Integral Ring Homomorphism
• Tensor with M
• The Tensor of 2 Integral Algebras
• Lifting Prime Ideals, Going Up, Going Down
• Integral Closure in the Polynomial Ring
• Extending a Ring Homomorphism

Valuation Rings

• Valuation Ring
• Local and Integrally Closed
• Linearly Ordered Ideals
• Dominant Local Ring
• Dominant Homomorphism
• Valuation Ring iff Maximal Dominant Local Ring
• Valuation Group
• The Valuation of the Sum
• From Valuation Group to Valuation Ring
• Ideals, PID, DVR, Noetherian
• Any Group will Do
• Valuation Metric
• Completing the Valuation Metric Space
• Algebraic p-adic Numbers
• p-adic is Locally Compact
• The Completion of the Localization
• Canonical Series
• Extending the Valuation to the Completion

Dedekind Domains

• Unique Factorization of Ideals
• Quotient and Localization
• A Nonzero Prime Ideal is Maximal
• Fractional Ideal
• A Homomorphism on a Fractional Ideal
• Invertible Ideal
• Unique Factorization of Fractional Ideals
• Dedekind implies Noetherian
• Invertible Ideal and Localization
• Localization Produces a DVR
• Integrally Closed and One Prime Ideal
• Invertible iff Projective
• 11 Definitions
• Fraction Ring is Dedekind
• Extending a Dedekind Domain
• UFD iff PID
• Finitely Many Prime Ideals
• To Contain is to Divide
• Two Generators
• Class Group
• The Index of an Ideal
• Conditions for a Finite Class Group

Elliptic Curves

• Introduction
• Technical Definition
• Elliptic Group
• Continuous Group
• Anticurve
• Finite Fields
• Families of Curves
• When the Cubic Splits
• Norm and Antinorm
• The s Factor
• From Rationals to Integers
• Elliptic Curve Factoring
• Elliptic Curve Cryptography
• Elliptic Curves in Characteristic 2

Lattice in n Space

• Lattice in n Space
• Covolume
• Cell Count and Index
• Cell Count Second Proof
• Discriminant
• Shifted Lattice
• Cluster Point
• The Intersection of Two Lattices
• Minimal Nonzero Lattice Point
• Finitely Many Sublattices of a Given Index
• The Point Lattice Theorem
• Polynomial and Laurent Rings

Integral Rings

• Group of Ring Automorphisms
• Localization
• Localization and the Fixed Ring
• Base Change
• Automorphisms on an Integral Domain
• G Acts Transitively on Prime Ideals
• Integral Ring
• Integral over a UFD
• Trapped Between two Free Modules of rank n
• Integral Rings and Field Extensions Correspond
• Finite Index Propagates Upward
• Finite Ideal Count Propagates Upward
• Valuation Propagates Upward
• The Splitting Problem
• Localization and The Splitting Problem
• Ramification / Residue Degree
• Totally Ramified, Unramified
• A Tower of Extensions
• When the Extension is Galois
• A Lumpy Split
• Primes Move in Parallel Cycles
• Primes over Primes in an Integral Ring
• Finding Two Generators
• The Norm of a Principal Ideal
• The Norm of an Ideal
• Decomposition Group
• Frobenius Conjugacy Class

Algebraic Number Theory

• Number Fields
• Index and Norm
• Algebraic Integers are Dedekind
• Global Field has a Finite Class Group
• Number Field as Lattice
• The Geometry of Numbers
• Finitely Many Extensions per Discriminant
• The Dirichlet Unit Theorem
• The Arakelov Picard Group
• pic₀ is Compact
• The Dirichlet Unit Theorem Part 2
• The Regulator
• Alternate Formula for Discriminant
• Unit Discriminant and Integrally Closed

Quadratic Number Fields

• Adjoining the square root of d
• Familiar Examples
• Finding the Integral Ring
• Primes over Primes
• Finding the Fundamental Unit
• Sample UFDs
• The Cube Root of 2
• The Cube Root of 3

Continued Fractions

• Continued Fractions that approach r
• Performance of the gcd Algorithm
• Continued Fractions and the Fundamental Unit
• Periodic

Cyclotomic Number Fields

• Cyclotomic and Quadratic
• 1-y over n
• Integrally Closed and Dedekind
• Cyclotomic through a Prime
• Sample UFDs
• Cyclotomic through a Prime Power
• Cyclotomic through a Composite
• 1-y over Composite
• The Fundamental Unit in Cyclo 8

Generalized Reciprocity

• Extending Reciprocity
• Gaussian Quadratic Reciprocity
• Gaussian Quartic Reciprocity
• Eisenstein Quadratic Reciprocity
• Cubic Reciprocity
• Quintic Reciprocity
• The Norm Jacobi Principle
• Jacobi and the Fixed Ring
• The Jacobi Prime Test
• The Strong Pseudoprime Test

Local Fields

• Complete DVR, Local Field
• All the Powers of p
• Completion = Inverse Limit
• Hensel's First Lemma
• Hensel's Second Lemma
• Hensel's Third Lemma
• The Chinese Remainder Theorem, Completion
• Completing a PID
• The Splitting Problem in the Completion
• Extending the Valuation of a CDVR
• Separable implies Simple
• Kernel of R[x] onto S