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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
• Algebraic number theory