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 $\mathrm(G)$ 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

:: $\mathfrak_ = \prod_\mathfrak(\chi)^,$

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

Example

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

::$\sum_^ (\varphi(p^i) - \varphi(p^))i = n\varphi(p^n) - 1 - (p-1)\sum_^p^i = n\varphi(p^n) - p^.$

Notes

References

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* {{Neukirch ANT

Algebraic number theory