Take K a number field and a divisor of K. A congruence subgroup H is defined as a subgroup of the group of all fractional ideals relative prime to m ( ) that contains all principal ideals that are generated by elements of K 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 H equals the kernel of the Artin map, then L is called the class field of H.

To formulate the main theorems, the equivalence relation on congruence subgroups is needed, namely that H 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 L. The classification theorem states that for each number field K, there is a unique one-to-one correspondence between the Abelian extensions and the equivalence classes of congruence subgroups H.

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.

 

Class Field, Class Number, Reciprocity Law