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 [E : K] = 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 extension OE (principal ideal theorem). Every prime ideal P of OK decomposes into the product of hK / f prime ideals in OE, where f is the order of [P] 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 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 . Notes References J. S. Milne, Class Field Theory (Course notes available at http://www.jmilne.org/math/). See the Introduction chapter of the notes, especially p. 4. Class field theory