Class Field Theory

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.

REFERENCES:

Garbanati, D. "Class Field Theory Summarized." Rocky Mtn. J. Math. 11, 195-225, 1981.

Hazewinkel, M. "Local Class Field Theory is Easy." Adv. Math. 18, 148-181, 1975.