Class Field

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 .

REFERENCES:

Cohn, H. Introduction to the Construction of Class Fields. New York: Cambridge University Press, 1985.