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

Mathematics-related lists