Conductor-discriminant formula

In mathematics, the conductor-discriminant formula or Führerdiskriminantenproduktformel, introduced by for abelian extensions and by   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 $$\mathrm{Irr}(G)$$ of the Galois group $$G = G(L/K)$$.

Statement
Let $$L/K$$ be a finite Galois extension of global fields with Galois group $$G$$. Then the discriminant equals


 * $$\mathfrak{d}_{L/K} = \prod_{\chi \in \mathrm{Irr}(G)}\mathfrak{f}(\chi)^{\chi(1)},$$

where $$\mathfrak{f}(\chi)$$ equals the global Artin conductor of $$\chi$$.

Example
Let $$L = \mathbf{Q}(\zeta_{p^n})/\mathbf{Q}$$ be a cyclotomic extension of the rationals. The Galois group $$G$$ equals $$(\mathbf{Z}/p^n)^\times$$. Because $$(p)$$ is the only finite prime ramified, the global Artin conductor $$\mathfrak{f}(\chi)$$ equals the local one $$\mathfrak{f}_{(p)}(\chi)$$. Because $$G$$ is abelian, every non-trivial irreducible character $$\chi$$ is of degree $$1 = \chi(1)$$. Then, the local Artin conductor of $$\chi$$ equals the conductor of the $$\mathfrak{p}$$-adic completion of $$L^\chi = L^{\mathrm{ker}(\chi)}/\mathbf{Q}$$, i.e. $$(p)^{n_p}$$, where $$n_p$$ is the smallest natural number such that $$U_{\mathbf{Q}_p}^{(n_p)} \subseteq N_{L^\chi_\mathfrak{p}/\mathbf{Q}_p}(U_{L^\chi_\mathfrak{p}})$$. If $$p > 2$$, the Galois group $$G(L_\mathfrak{p}/\mathbf{Q}_p) = G(L/\mathbf{Q}_p) = (\mathbf{Z}/p^n)^\times$$ is cyclic of order $$\varphi(p^n)$$, and by local class field theory and using that $$U_{\mathbf{Q}_p}/U^{(k)}_{\mathbf{Q}_p} = (\mathbf{Z}/p^k)^\times$$ one sees easily that if $$\chi$$ factors through a primitive character of $$(\mathbf{Z}/p^i)^\times$$, then $$\mathfrak{f}_{(p)}(\chi) = p^i$$ whence as there are $$\varphi(p^i) - \varphi(p^{i-1}) $$ primitive characters of $$(\mathbf{Z}/p^i)^\times$$ we obtain from the formula $$\mathfrak{d}_{L/\mathbf{Q}} = (p^{\varphi(p^n)(n - 1/(p-1))})$$, the exponent is


 * $$ \sum_{i = 0}^{n} (\varphi(p^i) - \varphi(p^{i-1}))i = n\varphi(p^n) - 1 - (p-1)\sum_{i=0}^{n-2}p^i = n\varphi(p^n) - p^{n-1}.$$