A class field for a set 
of primes, in the splitting-prime
formulation of Cohn (1985, p. 12), is a number field 
that is a maximal normal
extension of the rationals in which all the
primes in 
split completely, with 
the maximal set of primes
split by 
.
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 
(Cohn 1985, pp. 12-13). Every
such prime is expressible as the sum
of two squares, giving the factorization 
.
A sequence obtained by repeatedly passing from a number field to its Hilbert class field is called a class field tower.
