Hilbert class field
From Wikipedia, the free encyclopedia
In algebraic number theory, the Hilbert class field E of a number field K. is the maximal abelian unramified extension of K.
Note that in this context, 'unramified' is meant not only for the finite places (the classical ideal theoretic interpretation) but also for the infinite places. That is, every real embedding of K extends to a real embedding of E. As an example of why this is necessary, consider some real quadratic field.
The existence of unique Hilbert class field for given number field K was conjectured by David Hilbert and proved by Phillip Furtwängler. The existence of the Hilbert class field is a valuable tool in studying the structure of the ideal class group of given field.
[edit] Additional properties
Furthermore, E satisfies the following:
- E is a finite extension of K and [E:K] = hK, where hK is the class number of K.
- E is Galois over K.
- The ideal class group of K is isomorphic to the Galois group of E over K.
- Every ideal of
is a principal ideal of the ring extension 
- Every prime ideal
of decomposes into the product of 
prime ideals in 
, where f is the order of ![[\mathcal{P}]](../../../math/3/9/e/39e589ff1acbeb28fa0e3a072ee35b54.png)
in the ideal class group of
In fact, E is the unique field satisfying the above five properties.
This article incorporates material from Existence of Hilbert class field on PlanetMath, which is licensed under the GFDL.
