Hilbert class field

The Hilbert class field of a number field is its maximal abelian extension unramified at all finite primes. Its Galois group over the field is canonically isomorphic to the ideal class group.

Artin reciprocity: A reciprocity law connecting abelian Galois groups of number fields with ideal or idèle class groups. It identifies the Galois group of the Hilbert class field with the ideal class group.
Number field: A finite extension of the rational number field. The Hilbert class field is defined relative to a number field.
Quadratic field: A degree-two extension of the rational number field. Quadratic fields provide the simplest settings for explicit Hilbert class field calculations.
Narrow Hilbert class field: The maximal abelian extension unramified at finite primes and split completely at real places, corresponding to the narrow ideal class group. It imposes a real-place condition absent from the ordinary Hilbert class field.
Ideal class group: The finite abelian group of nonzero fractional ideals modulo principal fractional ideals in a number field. Its elements correspond exactly to automorphisms of the Hilbert class field.
Ring of integers: The integral closure of the integers in a number field. Its ideals supply the arithmetic objects used in the class group and Artin map.
Class number one: The property of a number field whose ideal class group is trivial. Such a field has no nontrivial Hilbert class field extension.
Ray class field: An abelian extension associated by class field theory with a modulus imposing congruence and ramification conditions. Ray class fields allow specified ramification, whereas the Hilbert class field is unramified at finite primes.
Frobenius element: A Galois group element associated with an unramified prime, acting on residue fields by raising elements to the residue-field cardinality. Prime ideals determine automorphisms through their Frobenius elements under the Artin map.
Dedekind domain: A Noetherian integrally closed domain of Krull dimension one, where nonzero ideals factor uniquely into prime ideals. The ring of integers is a Dedekind domain, enabling ideal-theoretic class field constructions.

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