A Hilbert class field 
of a number field 
is the unique maximal unramified Abelian
extension of 
which contains all other unramified Abelian extensions of 
. The field 
is therefore a finite extension of 
whose 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 
.
Iterating the construction of Hilbert class fields gives a class field tower.
