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), P = {p:p congruent 1 (mod 4)}. The class field for P is Q(i) because every such prime is expressible as the sum of two squares p = x^2 + y^2 = (x + i y)(x - i y).

class number | Hilbert class field | ideal | ideal extension | local class field theory | prime ideal | unique factorization