Conductor-discriminant formula

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 ${\displaystyle L/K}$ of local or global fields from the Artin conductors of the irreducible characters ${\displaystyle \mathrm {Irr} (G)}$ of the Galois group ${\displaystyle G=G(L/K)}$.

Statement

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

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

where ${\displaystyle {\mathfrak {f}}(\chi )}$ equals the global Artin conductor of ${\displaystyle \chi }$.^[1]

Example

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

${\displaystyle \sum _{i=0}^{n}(\varphi (p^{i})-\varphi (p^{i-1}))i=n\varphi (p^{n})-1-(p-1)\sum _{i=0}^{n-2}p^{i}=n\varphi (p^{n})-p^{n-1}.}$