Conductor-discriminant formula

In mathematics, the conductor-discriminant formula or Führerdiskriminantenproduktformel, introduced by Template:Harvs for abelian extensions and by Template:Harvs 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 $\mathrm{I}\mathrm{r}\mathrm{r}(G)$ 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

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

where $𝔣(\chi )$ equals the global Artin conductor of $\chi$.Template:Sfn

Example

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

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

References

• Template:Citation
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• Template:Neukirch ANT