Conductor of an Abelian extension

Let be an Abelian extension and let be the corresponding subgroup of the idèle class group (cf. Class field theory). The conductor of an Abelian extension is the greatest common divisor of all positive divisors such that is contained in the ray class field .

For an Abelian extension of local fields the conductor of is , where is the maximal ideal of (the ring of integers of) and is the smallest integer such that , . (Thus, an Abelian extension is unramified if and only if its conductor is .) The link between the local and global notion of a conductor of an Abelian extension is given by the theorem that the conductor of an Abelian extension of number fields is equal to , where is the conductor of the corresponding local extension . Here for the infinite primes, or according to whether or .

The conductor ramification theorem of class field theory says that if is the conductor of a class field , then is not divisible by any prime divisor which is unramified for and is divisible by any prime divisor that does ramify for .

If is the cyclic extension of a local field with finite or algebraically closed residue field defined by a character of degree 1 of , then the conductor of is equal to , where is the Artin conductor of the character (cf. Conductor of a character). Here is the separable algebraic closure of . There is no such interpretation known for characters of higher degree.

References

[a1]	J.-P. Serre, "Local fields" , Springer (1979) (Translated from French)
[a2]	J. Neukirch, "Class field theory" , Springer (1986) pp. Chapt. 4, Sect. 8