In mathematics, the conductor-discriminant formula or Führerdiskriminantenproduktformel, introduced by Hasse (1926, 1930) for abelian extensions and by Artin (1931) 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
be a finite Galois extension of global fields with Galois group
Then the discriminant equals where
equals the global Artin conductor of
χ
be a cyclotomic extension of the rationals.
The Galois group
is the only finite prime ramified, the global Artin conductor
( χ )
equals the local one
( χ )
is abelian, every non-trivial irreducible character
χ
1 = χ ( 1 )
Then, the local Artin conductor of
χ
equals the conductor of the
χ
( χ )
is the smallest natural number such that
χ
, 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
primitive characters of
we obtain from the formula