Conductor-discriminant formula explained

In mathematics, the conductor-discriminant formula or Führerdiskriminantenproduktformel, introduced by for abelian extensions and by for Galois extensions, is a formula calculating the relative discriminant of a finite Galois extension

of local or global fields from the Artin conductors of the irreducible characters of the Galois group .

Statement

Let

be a finite Galois extension of global fields with Galois group . Then the discriminant equals

where

equals the global Artin conductor of .

Example

Let

be a cyclotomic extension of the rationals. The Galois group equals . Because is the only finite prime ramified, the global Artin conductor equals the local one . Because is abelian, every non-trivial irreducible character is of degree . Then, the local Artin conductor of equals the conductor of the -adic completion of , i.e. , where is the smallest natural number such that }). If , the Galois group is cyclic of order , and by local class field theory and using that one sees easily that if factors through a primitive character of , then whence as there are primitive characters of we obtain from the formula , the exponent is