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
Read more about this topic: Conductor-discriminant Formula
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