Given a number field 
, there exists a unique maximal unramified Abelian
extension 
of 
which contains all other unramified Abelian extensions of 
. This finite field extension 
is called the Hilbert class field of 
. By a theorem of class field theory, the Galois
group 
is isomorphic to the class group of 
and for every subgroup 
of 
, there exists a unique unramified Abelian extension 
of 
contained in 
such that 
.
The degree ![[L:K]](/images/equations/HilbertClassField/Inline15.svg)
of 
over 
is equal to the class number of 
.
