A class field tower of a number field 
is the sequence of number fields
 |
in which 
is the Hilbert class field of 
. The tower is said to be finite if the sequence eventually
stabilizes, meaning that 
for all sufficiently large 
, and infinite otherwise.
The Golod-Shafarevich theorem implies that infinite class field towers exist (Golod and Shafarevich 1964).
