Given a set P of primes, a field K is called a class field if it is a maximal normal extension of the rationals which splits all of the primes in P, and if P is the maximal set of primes split by K. Here the set P 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 P is because every such prime is expressible as the sum of two squares .

 

Class Number, Hilbert Class Field, Ideal, Ideal Extension, Local Class Field Theory, Prime Ideal, Unique Factorization