Let of discriminant. The field has discriminant and so is an everywhere unramified extension of K, and it is abelian. Using the Minkowski bound, one can show that K has class number 2. Hence, its Hilbert class field is. A non-principal ideal of K is, and in L this becomes the principal ideal.
To see why ramification at the archimedean primes must be taken into account, consider the real quadratic fieldK obtained by adjoining the square root of 3 to Q. This field has class number 1 and discriminant 12, but the extension K/K of discriminant 9=32 is unramified at all prime ideals in K, so K admits finite abelian extensions of degree greater than 1 in which all finite primes of K are unramified. This doesn't contradict the Hilbert class field of K being K itself: every proper finite abelian extension of K must ramify at some place, and in the extension K/K there is ramification at the archimedean places: the real embeddings of K extend to complex embeddings of K.
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 =hK, where hK 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 OK extends to a principal ideal of the ring extensionOE.
Every prime idealP of OK decomposes into the product of hK/f prime ideals in OE, where f is the order of in the ideal class group of OK.
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 fieldwith 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.
