In algebraic number theory, the Hilbert class field E of a number field K is the maximal abelian unramified extension of K. Its degree over K equals the class number of K and the Galois group of E over K is canonically isomorphic to the ideal class group of K using Frobenius elements for prime ideals in K.
In this context, the Hilbert class field of K is not just unramified at the finite places (the classical ideal theoretic interpretation) but also at the infinite places of K. That is, every real embedding of K extends to a real embedding of E (rather than to a complex embedding of E).
The existence of a (narrow) Hilbert class field for a given number field K was conjectured by David Hilbert (1902) and proved by Philipp Furtwängler. The existence of the Hilbert class field is a valuable tool in studying the structure of the ideal class group of a given field.
The Hilbert class field E also satisfies the following:
In fact, E is the unique field satisfying the first, second, and fourth properties.
If K is imaginary quadratic and A is an elliptic curve with complex multiplication by the ring of integers of K, then adjoining the j-invariant of A to K gives the Hilbert class field.
In class field theory, one studies the ray class field with respect to a given modulus, which is a formal product of prime ideals (including, possibly, archimedean ones). The ray class field is the maximal abelian extension unramified outside the primes dividing the modulus and satisfying a particular ramification condition at the primes dividing the modulus. The Hilbert class field is then the ray class field with respect to the trivial modulus 1.
The narrow class field is the ray class field with respect to the modulus consisting of all infinite primes. For example, the argument above shows that is the narrow class field of .
This article incorporates material from Existence of Hilbert class field on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.