Linked from

The 25 pages that link to Algebraic number field, each with the reason it gives.

• Algebraic number theoryBroader topic: Its elements provide the arithmetic setting that the subject studies.

• Cyclotomic fieldNarrower topic: Every cyclotomic field is an algebraic number field.

• Ring of integersNarrower topic: The ring of integers is defined separately for each such field.

• Algebraic closureRelated: Its algebraic closure contains every algebraic number and supplies a common setting for number-field extensions.

• Ideal class groupNarrower topic: Class groups of number fields encode arithmetic in their rings of integers.

• Dedekind domainRelated: Its ring of integers is a central source of Dedekind domains.

• Dedekind zeta functionNarrower topic: Each number field has its own Dedekind zeta function.

• Algebraic integerNarrower topic: Algebraic integers in a number field form its ring of integers.

• Dirichlet's unit theoremNarrower topic: The theorem applies to the integers inside each such extension.

• Helmut HasseRelated: Hasse studied number fields by examining their completions at different primes.

• Hilbert class fieldNarrower topic: The Hilbert class field is defined relative to a number field.

• Local-global principleNarrower topic: Number fields are standard global settings whose completions supply arithmetic local fields.

• Arithmetic geometryRelated: Number fields provide central domains over which arithmetic varieties are studied.

• Kronecker–Weber theoremRelated: Every finite extension considered by the theorem is an algebraic number field.

• Mordell–Weil theoremNarrower topic: The theorem’s base field must have finite degree over the rationals.

• DiscriminantNarrower topic: Its ring of integers and trace pairing define the number-field discriminant.

• Class number formulaNarrower topic: The formula applies to this field and its arithmetic invariants.

• Hasse–Minkowski theoremRelated: It supplies the global field whose completions are tested.

• Bombieri–Lang conjectureNarrower topic: The conjecture concerns rational points over these arithmetic fields.

• Manin conjectureRelated: It supplies the field over which rational points and heights are defined.

• Albert–Brauer–Hasse–Noether theoremNarrower topic: The theorem's global field is a number field.

• Birch–Tate conjectureNarrower topic: The conjecture is formulated separately for each number field.

• Grothendieck–Katz p-curvature conjectureNarrower topic: The equation’s coefficients lie in this arithmetic setting before reduction modulo primes.

• Leopoldt's conjectureNarrower topic: The conjecture ranges over every such field.

• Melanie WoodBroader topic: Number fields are among the arithmetic objects whose counts and statistical properties Wood studies.