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 L / K of local or global fields from the Artin conductors of the irreducible characters I r r ( G ) of the Galois group G = G ( L / K ) .
Let L / K be a finite Galois extension of global fields with Galois group G . Then the discriminant equals
where f ( χ ) equals the global Artin conductor of χ .
Let L = Q ( ζ p n ) / Q be a cyclotomic extension of the rationals. The Galois group G equals ( Z / p n ) × . Because ( p ) is the only finite prime ramified, the global Artin conductor f ( χ ) equals the local one f ( p ) ( χ ) . Because G is abelian, every non-trivial irreducible character χ is of degree 1 = χ ( 1 ) . Then, the local Artin conductor of χ equals the conductor of the p -adic completion of L χ = L k e r ( χ ) / Q , i.e. ( p ) n p , where n p is the smallest natural number such that U Q p ( n p ) ⊆ N L p χ / Q p ( U L p χ ) . If p > 2 , the Galois group G ( L p / Q p ) = G ( L / Q p ) = ( Z / p n ) × is cyclic of order φ ( p n ) , and by local class field theory and using that U Q p / U Q p ( k ) = ( Z / p k ) × one sees easily that f ( p ) ( χ ) = ( p φ ( p n ) ( n − 1 / ( p − 1 ) ) ) : the exponent is
