existence of Hilbert class field
, where is the class number of .
is Galois over .
Every ideal of is a principal ideal of the ring extension .
Every prime ideal of decomposes into the product of prime ideals in , where is the order (http://planetmath.org/Order) of in the ideal class group of .
There is a unique field satisfying the above five properties, and it is known as the Hilbert class field of .
The field may also be characterized as the maximal abelian unramified (http://planetmath.org/AbelianExtension) extension of . Note that in this context, the term ‘unramified’ is meant not only for the finite places (the classical ideal theoretic ) but also for the infinite places. That is, every real embedding of extends to a real embedding of . As an example of why this is necessary, consider some real quadratic field.
|Title||existence of Hilbert class field|
|Date of creation||2013-03-22 12:36:45|
|Last modified on||2013-03-22 12:36:45|
|Last modified by||mathcam (2727)|
|Defines||Hilbert class field|