Take K a number field and m a divisor of K. A congruence subgroup H is defined as a subgroup of the group of all fractional ideals relative prime to m (ℑ_K^m) that contains all principal ideals that are generated by elements of K that are equal to 1 (mod m). These principal ideals split completely in all Abelian extensions and are consequently part of the kernel of the Artin map for each Abelian extension L/K. When there exists an Abelian extension L/K such that m contains all the primes that ramify in L/K and such that H equals the kernel of the Artin map, then L is called the class field of H.

class field | class number | reciprocity theorem