Hilbert class field

In algebraic number theory, the Hilbert class field of a number field is the maximal abelian unramified extension of. Its degree over equals the class number of and the Galois group of over is canonically isomorphic to the ideal class group of using Frobenius elements for prime ideals in.
In this context, the Hilbert class field of is not just unramified at the finite places but also at the infinite places of. That is, every real embedding of extends to a real embedding of .

Examples

If the ring of integers of is a unique factorization domain, in particular if, then is its own Hilbert class field.
Let of discriminant. The field has discriminant and so is an everywhere unramified extension of, and it is abelian. Using the Minkowski bound, one can show that has class number 2. Hence, its Hilbert class field is. A non-principal ideal of is, and in this becomes the principal ideal.
The field has class number 3. Its Hilbert class field can be formed by adjoining a root of, which has discriminant −23.
To see why ramification at the archimedean primes must be taken into account, consider the real quadratic field obtained by adjoining the square root of 3 to. This field has class number 1 and discriminant 12. Its extension has discriminant, and so is unramified at all prime ideals in, which means that admits finite abelian extensions of degree greater than 1 in which all finite primes of are unramified. This doesn't contradict the Hilbert class field of being itself: every proper finite abelian extension of must ramify at some place, and in the extension there is ramification at the archimedean places: the real embeddings of extend to complex embeddings of.
Likewise, for, the extension has discriminant, so is unramified at all prime ideals in, but ramified at both real places.
By the theory of complex multiplication, the Hilbert class field of an imaginary quadratic field is generated by the value of the elliptic modular function at a generator for the ring of integers.

History

The existence of a Hilbert class field for a given number field K was conjectured by 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.

Additional properties

The Hilbert class field E also satisfies the following:

E is a finite Galois extension of K and = h_K, where h_K is the class number of K.
The ideal class group of K is isomorphic to the Galois group of E over K.
Every ideal of O_K extends to a principal ideal of the ring extension O_E.
Every prime ideal P of O_K decomposes into the product of h_K / f prime ideals in O_E, where f is the order of in the ideal class group of O_K.

In fact, E is the unique field satisfying the first, second, and fourth properties.

Explicit constructions

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.

Generalizations

In class field theory, one studies the ray class field with respect to a given modulus, which is a formal product of prime ideals. 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.