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