List of algebraic number theory topics

This is a list of algebraic number theory topics.

## Basic topics

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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

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

## General aspects

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* 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

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* 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

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

## Arithmetic geometry

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