Conductor-discriminant formula

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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 .

Statement[edit]

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

where equals the global Artin conductor of .[1]

Example[edit]

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

Notes[edit]

  1. ^ Neukirch 1999, VII.11.9.

References[edit]