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 .