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
- Stark–Heegner theorem
- 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
