Site Map for the Math Reference Project
Math Reference Project, Site Map
Most of the theorems contained in this web site are accessible here.
They are organized by topic,
and are (mostly) presented in the order you would encounter them,
if you were stepping through the topic.
There are exceptions however.
Consider a simple example.
The area of the triangle is easily derived,
but the area of the hexagon has to wait for the pythagorean theorem.
I don't like forward references, so if you read through "plane geometry"
you will encounter the area of the triangle,
the pythagorean theorem, and the area of the hexagon, in that order.
However, in this directory,
it seems clearer to keep the theorems on area together.
Also, there are some one-time-only lemmas
that aren't presented here.
These are steps on the road to proving a larger theorem,
and they don't make sense out of context.
Besides, this list is long enough already!
Number Theory
Introduction
Properties of Addition and Multiplication
Euclid's GCD Algorithm
Diophantine Equations
Prime Numbers
Unique Factorization
Infinitely Many Primes
The Roots of the Integers are Irrational
phi(n) and sigma(n)
Mersenne Primes and Fermat Primes
Mobius Function
Gaussian Integers
Modular Mathematics
Introduction
Notation for Numbers or Polynomials mod m
Casting Out Nines
Fermat's Little Theorem
The Pseudoprime Test
Powers and Roots mod m
RSA Encryption
Chinese Remainder Theorem
Unique Factorization of Polynomials
Primitive Root
The Group of Units mod m
Proving p is Prime When p-1 is Factored
Discrete Logs
The Carmichael Numbers
Wilson's Theorem
Quadratic Residues
Quadratic Reciprocity
Applications of Quadratic Reciprocity
The Jacobe Symbol
Extending the Integers into the Complex Plane
Complex Extensions of Z
Conjugate Primes
Primes Over Primes
Pythagorean Triples
Pythagorean Triples with Coefficients
Representing n as a Sum of Squares
Adjoining the Sixth Roots of 1
Fermat's Last Theorem
Fermat's Last Theorem, n = 3
Fermat's Last Theorem, n = 4
Characters and Gaussian Sums
Introduction
Gaussian Sum
Plane Geometry
Introduction
Euclid's Postulates
Points and Lines
Angles
Congruent Triangles
Isosceles and Equilateral Triangles
Vertical Angles
Parallel Lines and Interior Angles
Angles of a Triangle Sum to 180 Degrees
Shapes, Angles, and Area
Convex Shapes
The Sum of the Interior Angles of a Polygon
Quadrilaterals and Parallelograms
Rectangles, Squares, Trapezoids, Etc
Regular Polygons
The Area of Some Simple Polygons
The Pythagorean Theorem
The Area of the Hexagon and Octagon
The Circumference and Area of a Circle
Chords and Tangents
Central and Inscribed Angles
Regular Tilings
Regular Tilings
Semiregular Tilings
Quadrilateral Tiles the Plane
Regular Polyhedra
Counting the Faces of a Polyhedron
The Dual of a Polyhedron
Platonic Solids
Archimedean Solids
Fair Dice
The Rotations of a Polyhedron
Combinatorics
Introduction
Permutations
Representing Permutations
Permutations With Repeated Elements
Truncated Permutations
Circular Permutations
Combinations
Pascal's Triangle
Binomial Theorem
Binomial Theorem and Differentiation
Factors of p in n!
Inclusion Exclusion
No-Match Permutations
The Marriage Problem
Generating Functions
Introduction
Counting Words in a Context Free Grammar
The Catalan Numbers
Polya's Enumeration Theorem
Counting Trees
Graph Theory
Introduction
Graph Representations
Regular and Complete Graphs
Graph Isomorphisms
Walks Paths and Trails
Connected Graphs, Components
Graceful Graphs
Trees
Spanning Tree, Greedy Algorithm
Steiner Points and Trees
Eulerean trails
Hamiltonian Cycles
Cycle Basis
Matroids
Intersection Graphs
Ramsey Numbers
Planar Graphs
Introduction
K5 and K3,3
Grinsberg's Formula
Euler's Formula
Sylvester's Problem
Five Color Theorem
Seven Color Theorem
Limits and Continuity
Limits and Continuity
Least Upper Bound
The Limit of a Function
Uniform Continuity
Composing Functions and Limits
The Limit of a Vector Function
Adding and Scaling Limits
Multiplying and Dividing Limits
Continuous Function Attains its Maximum
Continuous Function Becomes Uniform
Infinite Products
Infinite Products
The Product of Two Infinite Series
The Infinite Product of Infinite Series
Sequences and Series
Sequences and Series
The Terms Approach Zero
Increasing/Decreasing Sequences
A Bounded Increasing Sequence Converges
Sequence Domination
Absolute and Conditional Convergence
Absolute Convergence and Norms
Telescoping Series
Alternating Series
Geometric Series
Root Test, Ratio Test
Dirichlet Convergence Test
The Harmonic Series
The Integral Test
A Series of Uniform Functions
A Series of Integrals
Weierstrass m Test
Multidimensional Series
The zeta Function
Introductionn
zeta(2)
Odds That Two Numbers Are Coprime
The Dirichlet/Riemann zeta Function
The Line of Absolute Convergence
The Line of Conditional Convergence
Zeta is Analytic
The inverse of zeta
The gamma Function
The l Series
The Hurwitz zeta Function
The [Generalized] Riemann Hypothesis
Continued Fractions
Introductionn
Definition, Convergence, and Correspondence
Linear Algebra
Introduction
Independent Vectors
Basis
Rank and Dimension
Linear Transformations
Matrices
Matrix Operations
Matrix = Linear Transformation
The Inverse of a Matrix or Function
Dot Product
Perpendicular to a Plane
Orthogonal and Orthonormal
Gram Schmidt Process
Polynomial Approximations
Legendre Polynomials
Determinants
Determinant, a Recursive Definition
Determinant, a Permutation Definition
Swapping Rows
The Transpose Has the Same Determinant
Adding and Scaling Rows
Triangular Matrices
Operations on a Triangular Matrix
Gaussian Elimination
Computing Rank
Solving Simultaneous Linear Equations
The Diagonalization of a Matrix
Inverting a Matrix
Inverting an Orthogonal Matrix
Morphing One Basis into Another
Elementary Row Operations
Cramer's Rules
Rigid Rotations
The Volume of a Cell
The Shoelace Formula, Area of a Polygon
Cross Product
Vandermonde Matrix
Circulant Matrix
Eigen Vectors, Eigen Values, and the Characteristic Polynomial
Lattices
Introduction
Sublattice, Covolume, Index, and Discriminant
A Lattice has no Cluster Points
The Point Lattice Theorem
The Intersection of Two Lattices
Similar Matrices
Similar Matrices, A Change of Basis
Eigen Values and Vectors are Basis Invariant
Diagonalizable
Schur's Theorem
Trace and Norm
Hermitian Matrices and Operators
Normal Matrix, Unitarily Diagonalizable
Orthogonal Eigen Vectors
Real/Imaginary Eigen Values
Quadratic Forms and Conic Sections
Introduction
Conic Sections, Ellipse, Hyperbola, Parabola
The Ice Cream Cone Proof
A Parabolic Mirror
An Elliptical Mirror
A Hyperbolic Mirror
Quadratic Surfaces: Paraboloid, Hyperboloid, Ellipsoid, etc
Normalizing the Quadratic Form
A Quadratic Form with an Unchanging Sign
Extremal Values of a Quadratic Form on the Unit Sphere
Jordan Canonical Forms
Introduction
Block Diagonal Matrix
Direct Sum, Invariant, Complementary
Nilpotent Transformations
Jordan Canonical Form
Jordan Canonical Matrices are Class Representatives
Matrix Polynomials
Introduction
The Order of a Matrix
Cayley Hamilton
Spectral Radius
Linear Transforms
Transforms and Operators
Operating on Sequences
Eigen Functions and Hermitian Operators
Implementing a Transform via a Generalized Matrix
Orthogonal Functions
Differential Calculus
Introduction
Definition of Differentiation
Differentiable Implies Continuous
Adding and Scaling Derivatives
The Product Rule
The Quotient Rule
The Chain Rule
The Derivative of the Inverse Function
The Derivative of a Polynomial
The Derivative of xr
L'hopital's Rule
Local Properties of Functions
Strictly Increasing or Decreasing
Mean Value Theorem
Local Maximum and Minimum
Locally Invertible
Slope and Concavity
Concavity Test
Taylor Polynomials
Taylor Polynomials
Various Functions and Their Taylor Series
Calculus of Several Input Variables
Introduction
Directional/Partial Derivatives
Total Derivative
Continuous Partials and Differentiability
Multivariable Chain Rule
Level Curves and Surfaces
Extremal Points on a Surface and the Hessian Matrix
Lagrange Multipliers
Equal Mixed Partials
One Mixed Partial Implies the Other
Linear Equations in First Partials
Calculus of Several Input and Output Variables
Vector Functions
Generalized Chain Rule
Locally Invertible
Integral Calculus
Introduction
Riemann Sums and Nets
Upper and Lower Sums
Riemann Integral
Integrable Functions
Adding and Scaling Integrals
Continuous Functions are Integrable
Fundamental Theorem of Calculus
The Indefinite Integral
Mean Value Theorem for Integrals
Second Mean Value Theorem
Techniques of Integration
Integration by Substitution
Integration by Parts
Integration by Trig Substitution
Integration by Partial Fractions
Multidimensional Integrals
Nested Integrals
Integration Through the Jacobian
Integration in Polar Coordinates
Cylindrical Coordinates
Spherical Coordinates
The Area Under the Bell Curve
The Volume of a Simplex
The Volume of the Hypersphere
The Volume of the Spheroid
Log and Exponential Functions
Log Function
Exponential Function
Adding Logs
Radioactive Decay
Atmospheric Pressure
Compound Interest
Trig Functions
Sine and Cosine
Circumference of a Circle
Area of a Circle
The Area of Part of a Circle
More Trig Functions
Trigonometry
Angle Addition Formula
Double Angle Formula
Half Angle Formula
Law of Sines and Cosines
Measure Theory and Lebesgue Integration
Introduction
Sigma Algebra
Measurable Space
Monotone Convergence Theorem
Fatou's Lemma
Integrable Functions
Dominated Convergence Theorem
Lp Space
Holder's Inequality
Minkowski's Inequality
Paths in n-space
The Path Function
Velocity and Acceleration
Paths in Polar Coordinates
Arc Length
Speed and Arc Length
Arc Length in Polar Coordinates
Cardioid
Normal Vector, Osculating Plane
Curvature
Radius of Curvature
Kepler's Three Laws of Orbital Mechanics
Center of Mass
Introduction
Center of Geometry
Locating the Center of Mass
Force Acts Through the Center of Mass
Centroid
The Centroid of a Triangle
Pappus Theorem, Centroids, Volume, and Surface Area
Moments
Angular Momentum
Line Integrals and Conservative Fields
Line Integrals
Kinetic Energy and Line Integrals
Path Integral
Path Independence and Conservative Fields
Radial Forces are Conservative
Perpetual Energy
Green's Theorem
Equal Mixed Partials and Conservative Fields
The Gravity of a Sphere
Tunneling Through the Earth
The Electric Field Inside a Capacitor
Surface Integrals
Surface Area, Concepts and Formulas
Another Formula for Surface Area
Surface Area and Arc Length
Surfaces of Revolution
Surface Area of Ellipsoids/Spheroids
Surface[Normal] Integral
Divergence and Curl, and the Del Notation
The Divergence Theorem
Complex Variables and Differentiation/Integration
Introduction
Demoivre's Formula
Fractional Linear Transforms
Differentiation
Cauchy Riemann Condition
Contour Integral
Cauchy Goursat Theorem
The Integral of an Analytic Function
Cauchy's Integral Formula
An Analytic Function is Infinitely Differentiable
Morera's Theorem
Liouville's Theorem
An Analytic Function Has No Local Maximum
The Fundamental Theorem of Algebra
Complex Power Series
Introduction
Convergence of the Power Series
The Laurent Series
The Circle of Convergence
Unique Representation
Complex Exponentiation
Zeros and Poles
Holomorphic and Meromorphic Functions
Isolated Zeros
Multiplying and Dividing Power Series
Extending an Analytic Function
Conformal Maps
Continuous Binomial Theorem
Schwartz's Lemma
Ordinary Differential Equations
Introduction
Direct Integration
Linear Equations, First Order
The Bernoulli Equation
Existence and Uniqueness
The Dimensionality Theorem
Linear Equations with Constant Coefficients
Method of Separation
Difference Equations
Introduction
Existence and Uniqueness
The Sum of Consecutive nth Powers
The Golden Ratio and the Fibonacci Sequence
Primality and Factorization
Introduction
The Strong Pseudoprime Test
Factoring Polynomials Over a Finite Field
The ns Prime Test
The Mersenne Prime Test
The Pollard p-1 Method
The Pollard rho Method
Elliptic Curve Factoring
Quadratic Sieve
Fourier Series
Introduction
Trig Functions are Mutually Orthogonal
Defining the Fourier Series
Phase Shifting the Fourier Series
Music and Harmonics
The Square Wave
A Fourier Series for Any Continuous Function
Root Mean Square
Logic and Proofs
Introduction
Propositional Calculus
Set Theory
Introduction
Set Operations
Relations and Functions
Equivalence Relation
Onto and 1-1 Functions
Onto Map is a 1-1 Map
The Pigeonhole Principle
Indexed Sets, Product Sets
The Direct Sum of Pointed Sets
Axioms and Ordinals
The First Three Axioms
The Next Three Axioms
The Replacement Axiom
All Sets Could be Finite
The Infinite Axiom
The Foundation Axiom
Transitive Sets and the Successor Operation
Ordinals
The Ordinals are Well Ordered
Initial Segments
Ordinals Represent Well Ordered Sets
Ordinal Arithmetic
Finite Induction
The Recursion Theorem and Transfinite Induction
Cardinality
Cardinality
The Cardinality of Finite Sets
Countable and Uncountable Sets
The Cardinality of the Reals
The Schroder Bernstein Theorem
Cardinal Numbers
The Cardinals do not Form a Set
Cardinals can be Bounded Without Power Set
Cantor Diagonalization
The aleph() Function
The Well Ordering Principle
Zorn's Lemma
The Continuum Hypothesis
Knaster's Fixed Point Theorem
Point Set Topology
Open and Closed Sets
Interior and Closure
Subspace Topology
Dense, Separable
A Base for the Topology
A Local Base for the Topology
First and Second Countable
Discrete and Indiscrete Topologies
The Order Topology
Separation Axioms [Hausdorff]
Urysohn's Lemma
The Limit of a Sequence
Continuous Functions
Locally Finite
Pasting Continuous Functions Together
Homeomorphisms
Product Space, Product Topology
The Direct Sum of Pointed Spaces
Connected and Path Connected
Simply Connected
Sewing Spaces Together - The Klein Bottle
Quotient Space
Compact Sets
Introduction
Compact and Hausdorff Spaces
Countably/Sequentially Compact
Semicontinuous and Uniform Convergence
The Compact Product Topology
Closed and Bounded in Rn
The Tikhonov Product Theorem
Locally Compact
Compactification
Compact Open Topology
Metric Spaces
Introduction
The Distance Metric
The Open Ball Topology
Alternate Bases
The Product of Metric Spaces
Cauchy Schwarz Inequality, Triangular Inequality
Generalized Euclidean Space
Bounded Sets, Diameter
Continuous Functions on Metric Spaces
Uniform Continuity
Notation for Common Metric Spaces
Distance, Isometry, Equivalent
The Countable Product of Metric Spaces is Metrizable
Urysonh's Metrization Criteria
Complete Metric Spaces
Cauchy Sequences
The Reals are Cauchy Sequences of Rationals
Decimal Representations of Real Numbers
Least Upper Bound
Dedekind Cuts
The Reals Form a Complete Metric Space
The Intermediate Value Theorem
The Completion of a Metric Space
The Lipschitz Constant
The Lipschitz Constant and Derivatives
Contraction Maps and Fixed Point Iteration
Baire Category Theorem
First and Second Category
Uniform Boundedness Principle
Cantor Set
Banach Spaces
Introduction
Normed Vector Space
Bounded Linear Operator
The Hahn Banach Theorem
Continuous Implies Bicontinuous
Closed Graph Theorem
Topological Vector Space
Finite Dimensional is Euclidean
Hilbert Space
Dot Product is Continuous
Characterizing the Separable Hilbert Space
Bounded Linear Operator is Dot Product
One Hilbert Space for Each Cardinal
Stone Weierstrass
Introduction
Properties of c(s)
The Stone Weierstrass Theorem; All Continuous Functions are Accessible
Maximal Ideals in c(s)
Simplexes and Complexes
Introduction
Barycentric Coordinates
Simplicial Maps
Star Convex
The Simplicial Approximation Theorem
Irreducible and Noetherian Spaces
Introduction
Generic Spaces
Zariski Spaces
Algebraic Topology
Introduction
Homotopic Functions and Homotopy Classes
The Homotopy Classes of the Circle
The Fundamental Group
The Fundamental Group of the Torus
The Fundamental Group of the Sphere
A Nonabelian Fundamental Group
The Fundamental Theorem of Algebra
Brouwer's Fixed Point Theorem, 2 Dimensions
Covering Spaces
Introduction
The Path Lifting Theorem
The Homotopy Lifting Theorem
The Covering Space Induces a Group Monomorphism
Lifting Criteria
Universal Covering Space
The Hawaiian Earring
The Automorphism Group of a Covering Space
Winding Number and Jordan Curves
Introduction
The Jordan Curve Theorem
Space Filling Curves
Knots in 3 Space
Singular Homology and Cohomology
Introduction
The Homology of a Star Convex Space
Homotopic Functions induce Identical Homology Homomorphisms
The Method of Acyclic Models
Singular Subdivisions
The Excision Theorem
The Homology of the Spheres
Brouwer's Fixed Point Theorem, n Dimensions
Excisive Couple
CW Complex
Group Theory
Introduction
Cancellation and Order
Subgroups and Cosets
Joining and Intersecting Subgroups
The Index of a Subgroup
Kernels and Quotient Groups
Correspondence Theorems
Group Homomorphisms and Isomorphisms
Endomorphisms and Automorphisms
Inner Automorphisms
Direct and Semidirect Products
Center, Centralizer, Normalizer
Dihedral and General Linear Groups
Coliniations
Even and Odd Permutations
Symmetric and Alternating Groups
Group Actions
Group Actions on Sets
Orbits and Stabilizers
Transitive Action, Doubly Transitive
Translation and Conjugation, Groups Acting on Themselves
Macay's Theorem
Fixed Point Principle
Strong Cayley Theorem
p Groups and Sylow Subgroups
The Congruent Index Principle
First Sylow Theorem
Second Sylow Theorem
Third Sylow Theorem
The Burnside Counting Theorem
The Burnside Polya Theorem
Colored Necklaces
Some Familiar Finite Groups
The Sum of the Elements in an Abelian Group
Torsion and Locally Finite Groups
Groups of Order p2 3
Classifying pq Groups
Simple Groups Consist of Even Permutations
Alternating Groups are Simple
p Cycles Generate an Alternating/Symmetric Group
The Simple Group of Order 168
Simple Groups of Low Order
Free Groups
Introduction and Definition
Free Groups are Free
Relations and Presentations
Free Abelian Groups
Schreier Nielsen Rewriting
The Subgroup of a Free Group is Free
Chains of Subgroups - Solvable and Nilpotent Groups
Normal Series, Solvable Group
The Subgroup and Factor Group of a Solvable Group
The Composition Series
Commutator Subgroup and Abelianization
Commutator Subgroup and Solvability
The Central Series, and Nilpotent Groups
Intersecting the Center with Normal Subgroups
Classifying Finite Nilpotent Groups
The Intersection of Maximal Subgroups is Nilpotent
The Zassenhaus Formula for Subgroups
Jordan Holder for Groups
Uniserial Groups
Krull Schmidt Decomposition
Ring Theory
Introduction
Units and Associates in a Ring
Integral Domain
The Definition of a [Left/Right/General] Ideal
Ring Homomorphisms
Chinese Remainder Theorem
Irreducible and Prime Elements
Maximal and Principal Ideals
Maximal Ideals and Fields
Prime Ideals
Prime Ideal Correspondence
Intersection and Prime Ideals
The Union of Prime Ideals
Semiprime Ideals
Prime and Semiprime Rings
A Prime Ideal Outside of a Multiplicative Set
Nilpotent and Idempotent Elements
The Extension and Contraction of an Ideal
Maximal Infinitely Generated Ideals
Formal Derivatives
Introduction
Repeated Roots
Weyl Algebra
Algebras
Introduction
The Tensor Product of Algebras
The Compositum of two Ring Extensions
Quadratic, Cubic, and Quartic Equations
Low Order Polynomials
Quadratic Equations
Cubic Equations
Quartic Equations
Modules
Introduction
Unitary Modules
SubModules
Module Homomorphisms
The Ring of Module Endomorphisms
A Basis for a Module
The Product of Dimensions
Finitely Generated and Presented
A Product of Rings
Conductor Ideal, Annihilator
Noetherian and Artinian Modules
Chains of Modules
Chains of Sets
Finitely Generated = Noetherian
Finitely Generated Prime Ideals
The Quotient and Kernel of a Noetherian Module
Hilbert's Basis Theorem, Noetherian Polynomial Ring
Simple Modules
Introduction
Jordan Holder for Modules
Semisimple Modules
Submodules and Quotient Modules
Semisimple is a Direct Sum of Simple Modules
Semisimple Rings
Semisimple Rings are Noetherian
Semisimple Inheritance
Projective Modules
Introduction
Split Exact Sequences, Projective Modules, and Free Modules
The Dual of a Module
The Tensor Product as a Bilinear Map
Flat Modules
Finite Flat Modules
Faithful Modules
The Invariant Factor Theorem
The Exterior Power
The Support of a Module
Finitely Generated Modules over a PID
Introduction
The Rank of a Free Module is Well Defined
The Submodule of a Free Module is Free
The Structure of a Finitely Generated Module
Homology Groups/Modules
Introduction
[Short] Exact Sequences
The Short Five Lemma and the Five Lemma
The Serpent Lemma
Graded Modules
Relative Homology
Long Exact Sequence
Triple Homology Sequence
Chain Homotopic
Direct Limit Respects Homology and Exactness
Cohomology
Relative Cohomology
Tor and Ext
Introduction
Tor is Commutative
Direct Sum and Product over tor and ext
Dual m is Flat
Absolutely Flat
Quaternions
Introduction
Taking the Square Root of a Quaternion
Quaternion Groups, Quaternion Integers
The Quaternion GCD Algorithm
Splitting and Irreducibility
An Algorithm to Represent n as the Sum of Four Squares
The Quaternions mod p
Twisted Rings and Generalized Quaternions
Division Rings
Introduction
Every Finite Division Ring is a Field
Every Finite Multiplicative Subgroup of a Division Ring is Cyclic
Algebraic extensions of the Reals
Herstein's Lemma
Herstein's Little Theorem
Centrally Finite
Jacobson Radical and Jacobson Semisimple Rings
Five Definitions for the Jacobson Radical
Jac(r) Contains the Nil Ideals
Jacobson Semisimple
Semiprimary Ring
Artinian Implies Noetherian
Nakiama's Lemma
Maximal Ideals in an Artinian Ring
Semilocal Ring
Von Neumann Ring
Boolean Ring / Lattice
Jacobson ring, Hilbert Ring
Primitive and Semiprimitive Rings
The Density Theorem
Group Rings
Group Representations
Moshkey's Theorem
Subdirect Product and Subdirectly Irreducible
The Little Ideal
Radical Ideals and Nil Radicals
Nil Ideals and Radicals in a Commutative Ring
Nil/Nilpotent Ideals, and the Upper Nil Radical
Radical Ideals and Semiprime Ideals
Lower Nil Radical
Kothe's Conjecture
String Representable
Brown Macoy Radical
Spec r, the Zariski Topology
Introduction
Spec r is Compact
The Dimension of r and of spec r
Spec r for Fields, Dedekind Domains, and Polynomial Rings
The Spectrum of a Boolean Ring
Integral Domains
Introduction
Euclidean Domains
Principal Ideal Domains
A PID that is not a Euclidean Domain
Factorization Domains
Unique Factorization Domains
GCD and UFD
Gauss' Lemma
Bezout's Identity
Eisenstein's Criterion
Integral Extensions
Introduction
Units in an Integral Extension
Some Multiple of an Algebraic element is Integral
Finitely Generated = Integral
Integrally Closed
Integrally Closed is a Local Property
Primes over Primes in an Integral Extension
Going Up and Going Down
Integral Rings in a Field Extension
Introduction
The Ring of Integers Becomes Dedekind
The Splitting Problem
Ramification / Residue Degree
Fraction Rings and Localization
Fraction Rings
Ideal Correspondence in the Fraction Ring
Standard Notation for Localization
Fractional Linear Transforms
The Saturation of a Multiplicatively Closed Set
The Saturation of an Ideal
Mapping one Fraction Ring into Another
Localizing About the Non Zero Divisors
The Structure of r/s/t
The Intersection of Fraction Rings
Valuation Rings
Introduction
Valuation Rings are Local and Integrally Closed
Linearly Ordered Ideals
Valuation Group
The Valuation of the Sum
Going From the Valuation Group Back to the Ring
Discrete Valuation Ring
Any Group can be a Valuation Group
Valuation Metric, P-Adic Topology
Completion, and the P-Adic Numbers
Algebraic P-adic Numbers
From PID to P-adic Elements
Dedekind Domains
Introduction
Prime Ideals are Maximal in a Dedekind Domain
Fractional Ideals
Invertible Ideals
Unique Factorization of Fractional Ideals
Invertible Ideals are Finitely Generated
Invertible Ideals Remain Invertible in the Fraction Ring
Localization Produces a DVR
Integrally Closed and One Prime Ideal
Invertible iff Projective
11 Definitions for a Dedekind Domain
When is a Dedekind Domain a UFD?
To Contain is to Divide
Class Group
The Index of an Ideal
Primary Ideals
Introduction
Primary Decomposition
Associate Primes and First Uniqueness
Sph and Second Uniqueness
Laskerian Rings
A Nonlaskerian Ring
Primary Modules
Field Theory
Introduction
Field Homomorphisms are 1-1
Field Homomorphisms are Linearly Independent
Ordered Fields
Field Extensions
Math in a Field Extension
Adjoined Elements
Algebraic and Transcendental
Algebraically Independent
Splitting Field
Normal Extension
Conjugate Roots are Indistinguishable
Simple Extension
Closed Field
Finite Fields
Introduction
Cyclic Multiplicative Groups
Math in a Finite Field
A Unique Field of Order p
Unique Field of Order pn
The Automorphisms of a Finite field
Galois Extensions
Galois Extensions
Galois Groups
Fixed Fields, Group/Field Correspondence
Stable Field Extensions
Finite Fields Have Cyclic Galois Groups
Every Finite Group is Galois
Separable Field Extensions
Separable Extensions
Characteristic 0 Implies Separable
Finite Fields are Separable
Galois Implies Separable
The Separable Subfield
The Primitive Element Theorem
Purely Inseparable Elements and Extensions
An Example, Finite and Inseparable
The Fundamental Theorem of Algebra
Norm and Trace in a Field Extension
Natural Irrationality
Linearly Disjoint
The Compositum of two Field Extensions
Cyclotomic Field Extensions
Introduction
General Rings
Cyclotomic Extensions are Galois
Cyclotomic Polynomials, ζn
Cyclotomic Polynomials are Irreducible over Z
Ratio Units
The Fundamental Unit in ζ8
Solvable Field Extensions
Introduction
The Galois Group of a Polynomial
Cyclic and Solvable Extensions
Solvable Polynomials
An Unsolvable Quintic
Straightedge Compass Construction
Introduction to Construction
Rational Points can be Constructed
Constructible Points form a Field
A Tower of Quadratic Extensions
Trisecting the Angle
Constructing Regular n-gons
Algebraic Number Theory
Introduction
Algebraic Integers are Dedekind
Global Fields have a Finite Class Group
The Geometry of Numbers
The Dirichlet Unit Theorem
Quadratic Number Fields
Cyclotomic Number Fields
Local Fields
Hensel's Three Lemmas
Algebraic Geometry
Introduction
Algebraic Set
Nullstellensatz
Noether Normalization
The Maximal Ideals of the Integer Polynomials
Jacobson Semisimple Propagates Upward
Algebraic Varieties
Introduction
The Dimension of a Variety
Catenary Rings
Intersecting with a Hypersurface
Graded Rings and Modules
Projective Varieties
Probability
Probability Theory
Independent Events
The Monty Hall Dilemma
The Odds of Various Poker Hands
The Birthday Paradox
Bayes Theorem, Conditional Probability
Density and Distribution Functions
Steady State Probabilities in Markov Chains, such as Monopoly
Mean, Average, Expected Value, Variance, Standard Deviation
Mean, Mode, and Median
The Law of Large Numbers
Uniform Distribution
Binomial Distribution
Geometric Distribution
Poisson Distribution
Exponential Distribution
Normal Distribution and the erf Function
Least Squares Fit
Elliptic Curves
Introduction
The Elliptic Group
Elliptic Curves over Finite Fields
Elliptic Curves over the Rationals
Elliptic Curves in Characteristic 2
Formal Languages, Machines, Grammars, Regular Expressions
Formal Languages
Grammars
Grammar Types
Grounded and Connected
Finite State Machines
Regular Expressions
Regular Languages, Grammars, Expressions, and Finite State Machines
Moore and Meally Machines
Two-Way FSMs
The Pumping Lemma
Regular Languages and Closure
Myhill Nerode Classes
Context Free Grammars and Languages
Introduction
Canonical Forms and Ambiguity
Chomsky/Greibach Normal Form
Linear and Operator Grammars
Push Down Machines
The Pumping Lemma
Turing Machines
Introduction
Definition and Generalizations - How to Simulate a Computer
Bounded Turing Machines
Machine/Grammar Equivalence
The Two Stack PDM
Universal TM, Universal Language
Undecidability
Rice's Theorem
Complexity Theory
Introduction
Satisfiability
Three-Sat and Two-Sat
Resolution
Exponential Time
p and np Space
Adding an Oracle to a Turing Machine
NP Complete Problems
Introduction
The Graph Coloring Problem
Finding a Hamiltonian Circuit
Traveling Salesman Problem
Vertex Cover Problem
Integer Solutions to Simultaneous Inequalities
Nonnegative or Closest Lattice Point
Group Element Representation, or Shortest Solution of a Permutation Puzzle
Exact Cover
Bin Packing Problem
Category Theory
Introduction
Objects and Morphisms
Representable Category
Free Objects
Monic and Epic Morphisms
Projective and Injective Objects
Initial and Terminal Objects
Commutative Diagrams
Product and Coproduct
Limit and Colimit
Pushout and Pullback
Graded Category
Direct Limit, Inverse Limit
Functor
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