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 formula_1 of local or global fields from the Artin conductors of the irreducible characters formula_2 of the Galois group formula_3.
Statement.
Let formula_1 be a finite Galois extension of global fields with Galois group formula_5. Then the discriminant equals
 formula_6
where formula_7 equals the global Artin conductor of formula_8.
Example.
Let formula_9 be a cyclotomic extension of the rationals. The Galois group formula_5 equals formula_11. Because formula_12 is the only finite prime ramified, the global Artin conductor formula_7 equals the local one formula_14. Because formula_5 is abelian, every non-trivial irreducible character formula_8 is of degree formula_17. Then, the local Artin conductor of formula_8 equals the conductor of the formula_19-adic completion of formula_20, i.e. formula_21, where formula_22 is the smallest natural number such that formula_23. If formula_24, the Galois group formula_25 is cyclic of order formula_26, and by local class field theory and using that formula_27 one sees easily that if formula_8 factors through a primitive character of formula_29, then formula_30 whence as there are formula_31 primitive characters of formula_29 we obtain from the formula formula_33, the exponent is
formula_34