Take a number field and a divisor of . A congruence subgroup is defined as a subgroup of the group of all fractional ideals relative prime to () that contains all principal ideals that are generated by elements of that are equal to 1 (mod ). These principal ideals split completely in all Abelian extensions and are consequently part of the kernel of the Artin map for each Abelian extension .
When there exists an Abelian extension such that contains all the primes that ramify in and such that equals the kernel of the Artin map, then is called the class field of .
To formulate the main theorems, the equivalence relation on congruence subgroups is needed, namely that and are called equivalent if there exists a divisor such that .
Class field theory consists of two basic theorems. The existence theorem states that to every equivalence class of congruence subgroups, there belongs a class field . The classification theorem states that for each number field , there is a unique one-to-one correspondence between the Abelian extensions and the equivalence classes of congruence subgroups .
This is important because this means that all Abelian extensions of a number field can be found using a property that is completely determined within the number field itself.