Hilbert class field

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.

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