Analytic objects: ζ-functions, L-functions, and class number formula


Algebraic number field
Bases for number fields


Algebraic number field
Bases for number fields
Integral basis


Algebraic number field
Bases for number fields
Power basis


Regular representation, trace and discriminant

Example
Properties
Example with integral basis


Examples
Archimedean places
Non-Archimedean or ultrametric places
Prime ideals in OK


Algebraic number field
Places
Archimedean places


Algebraic number field
Places
Non-Archimedean or ultrametric places


Algebraic number field
Places
Prime ideals in OK


Algebraic number field
Places
Prime ideals in OK
Lying over theorem and places


An example
Dedekind discriminant theorem


Algebraic number field
Ramification
An example


Algebraic number field
Ramification
Dedekind discriminant theorem


Algebraic number field
Galois groups and Galois cohomology


Algebraic number field
Local-global principle

Local and global fields
Hasse principle
Adeles and ideles


Algebraic number field
Local-global principle
Local and global fields


Algebraic number field
Local-global principle
Hasse principle


Algebraic number field
Local-global principle
Adeles and ideles


Terminology and definition
Linear algebra


Notation and terminology
History and etymology
Number of terms and use of fractions
Proportions and percentage ratios
Reduction
Irrational ratios
Odds
Units
Triangular coordinates


Ratio
History and etymology
Euclid's definitions


Ratio
Number of terms and use of fractions


Meanings in different areas of mathematics


Meanings in different areas of mathematics

Prevalent meaning
Meaning in measure theory
Meaning in number theory
Meaning in graph theory
Meaning in topology
Meaning in algebra


Motivation
Equivalent characterizations
Case of rings
Case of fields
Notes
References
Sources


Characteristic (algebra)
Equivalent characterizations


Fields of characteristic zero
Fields of prime characteristic


Characteristic (algebra)
Case of fields
Fields of characteristic zero


Characteristic (algebra)
Case of fields
Fields of prime characteristic


Definition
Rational points on curves
Higher dimensions
Counting points over finite fields


Rational point
Rational points on curves


Rational point
Rational points on curves
Genus 0


Rational point
Rational points on curves
Genus 1


Rational point
Rational points on curves
Genus at least 2


Varieties with few rational points
Varieties with many rational points


Rational point
Higher dimensions
Varieties with few rational points


Rational point
Higher dimensions
Varieties with many rational points


Rational point
Counting points over finite fields


Definition
Examples
Some theorems
Completion
Topologically complete spaces
Alternatives and generalizations


Complete metric space
Topologically complete spaces


Complete metric space
Alternatives and generalizations


Coordinate lines/curves and planes/surfaces


Coordinate system
Common coordinate systems

Cylindrical and spherical coordinate systems


Coordinate system
Common coordinate systems
Number line


Coordinate system
Common coordinate systems
Cartesian coordinate system


Coordinate system
Common coordinate systems
Polar coordinate system


Cylindrical and spherical coordinate systems


Coordinate system
Common coordinate systems
Homogeneous coordinate system


Coordinate system
Common coordinate systems
Other commonly used systems


Coordinate system
Coordinates of geometric objects


Coordinate system
Orientation-based coordinates


Definitions
Examples
Taylor series
Abstract algebra and geometric notion
Applications


Rational function
Abstract algebra and geometric notion

Notion of a rational function on an algebraic variety


Rational function
Abstract algebra and geometric notion
Complex rational functions


Notion of a rational function on an algebraic variety


Large number notation
Number theory
Linear algebra
Algebra
Geometry
Integrable systems
Dynamical systems
Three dimensional geometry
Functional analysis
Classical logic
Set theory
Game theory
Proof theory
Rewriting systems
Lambda calculus
Graph theory
Computing


Canonical form
Examples
Large number notation


Canonical form
Examples
Integrable systems


Canonical form
Examples
Dynamical systems


Canonical form
Examples
Three dimensional geometry


Canonical form
Examples
Functional analysis


Canonical form
Examples
Rewriting systems


Notation and testing
Properties
Coprimality in sets
Coprimality in ring ideals
Probability of coprimality
Generating all coprime pairs
Applications
Generalizations


Coprime integers
Coprimality in ring ideals


Coprime integers
Probability of coprimality


Coprime integers
Generating all coprime pairs


Examples Uniqueness Applications Generalization


Transmission and the manuscript tradition


Greek mathematics
Archaic and Classical periods


Greek mathematics
Hellenistic and Roman periods


History
Geometric interpretation
Square roots of negative numbers


Imaginary number
Geometric interpretation


Imaginary number
Square roots of negative numbers


Vocabulary
Forms of fractions
Arithmetic with fractions
Fractions in abstract mathematics
Algebraic fractions
Radical expressions
Typographical variations
History
In formal education


Reciprocals and the "invisible denominator"


Fraction
Forms of fractions
Simple, common, or vulgar fractions


Fraction
Forms of fractions
Proper and improper fractions


Fraction
Forms of fractions
Decimal fractions and percentages


Fraction
Forms of fractions
Mixed numbers


Fraction
Forms of fractions
Historical notions

Egyptian fraction
Complex and compound fractions


Fraction
Forms of fractions
Historical notions
Egyptian fraction


Fraction
Forms of fractions
Historical notions
Complex and compound fractions


Converting between decimals and fractions


Fraction
Arithmetic with fractions
Equivalent fractions


Fraction
Arithmetic with fractions
Equivalent fractions
Simplifying (reducing) fractions


Fraction
Arithmetic with fractions
Comparing fractions


Fraction
Arithmetic with fractions
Addition


Fraction
Arithmetic with fractions
Addition
Adding unlike quantities


Fraction
Arithmetic with fractions
Subtraction


Fraction
Arithmetic with fractions
Multiplication

Multiplying a fraction by another fraction

Multiplying a fraction by a whole number


Multiplying a fraction by another fraction


Multiplying a fraction by a whole number


Fraction
Arithmetic with fractions
Multiplication
Multiplying mixed numbers


Fraction
Arithmetic with fractions
Division


Converting between decimals and fractions

Converting repeating decimals to fractions


Converting repeating decimals to fractions


Fraction
Fractions in abstract mathematics


Pedagogical tools
Documents for teachers


Fraction
In formal education
Pedagogical tools


Fraction
In formal education
Documents for teachers


Overview
Applications
Calculation
Properties
Probabilities and expected value
In commutative rings


Definition
Example
Coprime numbers
A geometric view


Greatest common divisor
Overview
Definition


Greatest common divisor
Overview
Example


Greatest common divisor
Overview
Coprime numbers


Greatest common divisor
Overview
A geometric view


Reducing fractions
Least common multiple


Greatest common divisor
Applications
Reducing fractions


Greatest common divisor
Applications
Least common multiple


Using prime factorizations
Euclid's algorithm
Euclidean algorithm
Lehmer's GCD algorithm
Binary GCD algorithm
Other methods
Complexity


Greatest common divisor
Calculation
Using prime factorizations


Greatest common divisor
Calculation
Euclid's algorithm


Greatest common divisor
Calculation
Euclidean algorithm


Greatest common divisor
Calculation
Lehmer's GCD algorithm


Greatest common divisor
Calculation
Binary GCD algorithm


Greatest common divisor
Calculation
Other methods


Greatest common divisor
Calculation
Complexity


Greatest common divisor
Probabilities and expected value


Greatest common divisor
In commutative rings


Terminology
Rational fractions
Irrational fractions


Etymology and nicknames
History
Geography
Demographics
Economy
Culture and contemporary life
Sports
National protected areas
Government
Crime
Education
Media
Transportation
Notable people
Sister cities


French–Spanish colonial era
United States territorial era
Battle of New Orleans
Port
Civil War–Reconstruction era
Jim Crow era
20th century
21st century


New Orleans
History
French–Spanish colonial era


New Orleans
History
United States territorial era


New Orleans
History
Battle of New Orleans


New Orleans
History
Civil War–Reconstruction era


Civil Rights movement
Drainage and flood control


New Orleans
History
20th century
Civil Rights movement


New Orleans
History
20th century
Drainage and flood control


Hurricane Katrina
Hurricane Rita
Post-disaster recovery
Hurricane Ida


New Orleans
History
21st century
Hurricane Katrina


New Orleans
History
21st century
Hurricane Rita


New Orleans
History
21st century
Post-disaster recovery


New Orleans
History
21st century
Hurricane Ida


Elevation
Cityscape
Climate
Threat from tropical cyclones


Historic and residential architecture
Tallest buildings


New Orleans
Geography
Cityscape
Historic and residential architecture


New Orleans
Geography
Cityscape
Tallest buildings


New Orleans
Geography
Threat from tropical cyclones


Race and ethnicity
Sexual orientation and gender identity
Religion


New Orleans
Demographics
Race and ethnicity


New Orleans
Demographics
Sexual orientation and gender identity


Port
Business
Tourist and convention business
Federal and military agencies


New Orleans
Economy
Tourist and convention business


New Orleans
Economy
Federal and military agencies


New Orleans
Culture and contemporary life

Tourism
Entertainment and performing arts
Cuisine
Dialect


New Orleans
Culture and contemporary life
Tourism


New Orleans
Culture and contemporary life
Entertainment and performing arts


New Orleans
Culture and contemporary life
Cuisine


New Orleans
Culture and contemporary life
Dialect


Colleges and universities
Primary and secondary schools
Libraries


New Orleans
Education
Colleges and universities


New Orleans
Education
Primary and secondary schools


Public transportation
Bicycling
Roads
Taxi service
Airports
Rail
Modal characteristics


New Orleans
Transportation
Public transportation


New Orleans
Transportation
Public transportation
Streetcars


New Orleans
Transportation
Public transportation
Buses


New Orleans
Transportation
Public transportation
Ferries


New Orleans
Transportation
Modal characteristics


In Euclidean geometry
Plane projective curves
Remarkable points of a plane curve
Analytic structure
Non-plane algebraic curves
Algebraic function fields
Complex curves and real surfaces
Singularities
Examples of curves


Algebraic curve
Remarkable points of a plane curve

Intersection with a line
Tangent at a point
Asymptotes
Singular points


Algebraic curve
Remarkable points of a plane curve
Intersection with a line


Algebraic curve
Remarkable points of a plane curve
Tangent at a point


Algebraic curve
Remarkable points of a plane curve
Asymptotes


Algebraic curve
Remarkable points of a plane curve
Singular points


Algebraic curve
Non-plane algebraic curves


Algebraic curve
Algebraic function fields


Algebraic curve
Complex curves and real surfaces


Algebraic curve
Complex curves and real surfaces
Compact Riemann surfaces


Algebraic curve
Singularities
Classification of singularities


Rational curves
Rational plane curves
Elliptic curves
Curves of genus greater than one


Algebraic curve
Examples of curves
Rational curves


Algebraic curve
Examples of curves
Rational plane curves


Algebraic curve
Examples of curves
Elliptic curves


Algebraic curve
Examples of curves
Curves of genus greater than one

Projective plane curves
Curves in product of projective lines


Algebraic curve
Examples of curves
Curves of genus greater than one
Projective plane curves


Algebraic curve
Examples of curves
Curves of genus greater than one
Curves in product of projective lines


Derivation and other types of word formation


Morphological derivation
Derivational patterns


Morphological derivation
Derivation and inflection


Characteristics
Occurrence
History
Applications
Precautions
Transportation
In popular culture
Notes
References
External links


Physical properties
Allotropes
Nuclear fission
Isotopes and nucleosynthesis
Decay heat and fission properties
Compounds and chemistry
Alloys


Plutonium
Characteristics
Physical properties


Plutonium
Characteristics
Nuclear fission


Plutonium
Characteristics
Isotopes and nucleosynthesis


Plutonium
Characteristics
Decay heat and fission properties


Plutonium
Characteristics
Compounds and chemistry


Plutonium
Characteristics
Compounds and chemistry
Electronic structure


Discovery
Early research
Production during the Manhattan Project
Trinity and Fat Man atomic bombs
Cold War use and waste
Medical experimentation


Plutonium
History
Production during the Manhattan Project


Plutonium
History
Trinity and Fat Man atomic bombs


Plutonium
History
Cold War use and waste


Plutonium
History
Medical experimentation


Explosives
Mixed oxide fuel
Power and heat source


Plutonium
Applications
Power and heat source


Toxicity
Marine toxicity
Criticality potential
Flammability


Plutonium
Precautions
Criticality potential