List of algebraic number theory topics

This is a list of algebraic number theory topics.

Basic topics
These topics are basic to the field, either as prototypical examples, or as basic objects of study.
 * Algebraic number field
 * Gaussian integer, Gaussian rational
 * Quadratic field
 * Cyclotomic field
 * Cubic field
 * Biquadratic field
 * Quadratic reciprocity
 * Ideal class group
 * Dirichlet's unit theorem
 * Discriminant of an algebraic number field
 * Ramification (mathematics)
 * Root of unity
 * Gaussian period

Important problems

 * Fermat's Last Theorem
 * Class number problem for imaginary quadratic fields
 * Stark–Heegner theorem
 * Heegner number
 * Langlands program

General aspects

 * Different ideal
 * Dedekind domain
 * Splitting of prime ideals in Galois extensions
 * Decomposition group
 * Inertia group
 * Frobenius automorphism
 * Chebotarev's density theorem
 * Totally real field
 * Local field
 * p-adic number
 * p-adic analysis
 * Adele ring
 * Idele group
 * Idele class group
 * Adelic algebraic group
 * Global field
 * Hasse principle
 * Hasse–Minkowski theorem
 * Galois module
 * Galois cohomology
 * Brauer group

Class field theory

 * Class field theory
 * Abelian extension
 * Kronecker–Weber theorem
 * Hilbert class field
 * Takagi existence theorem
 * Hasse norm theorem
 * Artin reciprocity
 * Local class field theory

Iwasawa theory

 * Iwasawa theory
 * Herbrand–Ribet theorem
 * Vandiver's conjecture
 * Stickelberger's theorem
 * Euler system
 * p-adic L-function

Arithmetic geometry

 * Arithmetic geometry
 * Complex multiplication
 * Abelian variety of CM-type
 * Chowla–Selberg formula
 * Hasse–Weil zeta function