Given a set 
of primes, a field 
is called a class field if it is a maximal normal
extension of the rationals which splits
all of the primes in 
,
and if 
is the maximal set of primes split by K. Here the set 
is defined up to the equivalence relation of allowing a finite
number of exceptions.
The basic example is the set of primes congruent to 1 (mod 4),
 |
The class field for 
is 
because every such prime is expressible
as the sum of two squares 
.
A sequence obtained by repeatedly passing from a number field to its Hilbert class field is called a class field tower.
