Number theory Algebraic number theory Concept
Algebraic number field
Summary
An algebraic number field, or number field, is a field extension of the rational numbers of finite degree. Number fields are the main objects of algebraic number theory. Each number field has a ring of integers , a Dedekind domain that plays the role of inside . The failure of unique factorisation in is measured by the ideal class group.
Where this leads
44 pages build on this one.
- Dirichlet's unit theorem
- Minkowski bound
- Dedekind domain
- Regulator of a number field
- Siegel's theorem on integral points
- Class number formula
- Idele class group
- Splitting of prime ideals
- Artin reciprocity law
- Quadratic field
- Ideal norm
- Class number
- Selmer group
- Class field theory
- Kronecker–Weber theorem
- Bombieri–Vinogradov theorem
- Cyclotomic field
- Vinogradov's theorem
- Ideal class group
- Generalized Riemann hypothesis
- Ramification
- Finiteness of the class number
- Faltings's theorem
- Hilbert class field
- Langlands program
- Tate–Shafarevich group
- Hermite–Minkowski theorem
- Hecke character
- Place (number field)
- Iwasawa theory
- Decomposition group
- Discriminant of a number field
- Frobenius element
- Adele ring
- Dedekind zeta function
- Fundamental identity for ramification and inertia
- Hardy–Littlewood circle method
- Global field
- Height function
- Siegel zero
- Chen's theorem
- Chebotarev's density theorem
- Siegel–Walfisz theorem
- Bounded gaps between primes
Further reading
About this page
| Tag | MWBDFS2KJ5 |
|---|---|
| URI | m2://MWBDFS2KJ5 |
| Kind | concept |
| Writing | W0 stub |
| MSC | 11R04, 11R21 |
| Specializes | Field, Global field |
| Same as | |
| Changed | 2 November 2026 |
Markdown · JSON · Edit on GitHub
What this page needs for W1
- a source with a locator (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.
What links here
Results about this
Used by
- Dirichlet's unit theorem
- Minkowski bound
- Dedekind domain
- Regulator of a number field
- Siegel's theorem on integral points
- Class number formula
- Idele class group
- Splitting of prime ideals
- Artin reciprocity law
- Quadratic field
- Ideal norm
- Class number
- Selmer group
- Class field theory
- Cyclotomic field
- Ideal class group
- Generalized Riemann hypothesis
- Ramification
- Finiteness of the class number
- Faltings's theorem
- Hilbert class field
- Langlands program
- Tate–Shafarevich group
- Hermite–Minkowski theorem
- Hecke character
- Place (number field)
- Iwasawa theory
- Decomposition group
- Discriminant of a number field
- Frobenius element
- Adele ring
- Dedekind zeta function
- Fundamental identity for ramification and inertia
- Global field
- Height function
- Chebotarev's density theorem