Hilbert class field

E459562

mathematical concept number field extension object in algebraic number theory

The Hilbert class field of a number field is its maximal unramified abelian extension, central in class field theory as it corresponds to the field’s ideal class group.

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Observed surface forms (1)

Surface form	Occurrences
Hilbert class fields of imaginary quadratic fields	1

Statements (49)

Predicate	Object
instanceOf	mathematical concept ⓘ number field extension ⓘ object in algebraic number theory ⓘ
appearsIn	Hilbert’s 12th problem NERFINISHED ⓘ theory of complex multiplication of elliptic curves ⓘ
characterizedBy	Artin reciprocity isomorphism with the ideal class group ⓘ conductor equal to the ring of integers of the base field ⓘ every ideal class of the base field becomes principal in it ⓘ no nontrivial unramified abelian extension of the base field lies strictly above it ⓘ
constructedVia	class field theory ⓘ idele class group and Artin map ⓘ ray class fields with trivial modulus ⓘ
contains	all finite unramified abelian extensions of the base field ⓘ
correspondsTo	ideal class group of the base field ⓘ maximal unramified abelian extension in class field theory ⓘ
definedOver	number field ⓘ
degreeOverBaseFieldEquals	class number of the base field ⓘ
fieldOfStudy	algebraic number theory ⓘ class field theory NERFINISHED ⓘ
forBaseField	number field with given ideal class group ⓘ
GaloisGroupIsomorphicTo	ideal class group of the base field ⓘ
generalizationOf	Hilbert class field of an imaginary quadratic field generated by j-invariants ⓘ
hasApplication	computing class numbers ⓘ studying capitulation of ideal classes in extensions ⓘ
hasProperty	Galois over the base number field ⓘ abelian over the base number field ⓘ finite extension of the base number field ⓘ its Galois group is isomorphic to the ideal class group of the base field ⓘ maximal unramified abelian extension of a number field ⓘ normal extension of the base number field ⓘ unique up to isomorphism for a given number field ⓘ unramified at all finite places of the base field ⓘ unramified at all finite primes of the base field ⓘ
hasSpecialCase	Hilbert class field of a real quadratic field ⓘ Hilbert class field of an imaginary quadratic field ⓘ Hilbert class field of the rational numbers NERFINISHED ⓘ
namedAfter	David Hilbert NERFINISHED ⓘ
relatedTo	Hilbert class polynomial NERFINISHED ⓘ Kronecker–Weber theorem in the case of Q NERFINISHED ⓘ maximal unramified extension ⓘ narrow Hilbert class field NERFINISHED ⓘ ray class field ⓘ
unramifiedAt	every nonzero prime ideal of the ring of integers of the base field ⓘ
usedIn	complex multiplication theory ⓘ construction of unramified extensions ⓘ explicit class field theory ⓘ proofs and formulations of class field theory ⓘ study of ideal class groups ⓘ study of the arithmetic of quadratic fields ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Kronecker–Weber theorem → relatedTo → Hilbert class field ⓘ

Gaussian periods → relatedTo → Hilbert class field ⓘ

this entity surface form: Hilbert class fields of imaginary quadratic fields