Conductor-discriminant formula


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 : the exponent is