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 $L/K$ 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

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

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 $L^{\chi }=L^{\mathrm{k}\mathrm{e}\mathrm{r}(\chi )}/\mathbf{Q}$, i.e. $(p{)}^{n_p}$, where $n_p$ is the smallest natural number such that $U_{{\mathbf{Q}}_p}^{(n_p)}\subseteq N_{L_{\mathfrak{𝔭}}^{\chi }/{\mathbf{Q}}_p}(U_{L_{\mathfrak{𝔭}}^{\chi }})$. 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 $\phi (p^n)$, and by local class field theory and using that $U_{{\mathbf{Q}}_p}/U_{{\mathbf{Q}}_p}^{(k)}=(\mathbf{Z}/p^k{)}^{\times }$ 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 $\phi (p^i)-\phi (p^{i-1})$ primitive characters of $(\mathbf{Z}/p^i{)}^{\times }$ we obtain from the formula ${\mathfrak{𝔡}}_{L/\mathbf{Q}}=(p^{\phi (p^n)(n-1/(p-1))})$, 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

• Artin, Emil (1931), "Die gruppentheoretische Struktur der Diskriminanten algebraischer Zahlkörper.", Journal für die Reine und Angewandte Mathematik (in German), 1931 (164): 1–11, doi:10.1515/crll.1931.164.1, ISSN 0075-4102, S2CID 117731518, Zbl 0001.00801{{citation}}: CS1 maint: unrecognized language (link)
• Hasse, H. (1926), "Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. I: Klassenkörpertheorie.", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 35: 1–55{{citation}}: CS1 maint: unrecognized language (link)
• Hasse, H. (1930), "Führer, Diskriminante und Verzweigungskörper relativ-Abelscher Zahlkörper.", Journal für die reine und angewandte Mathematik (in German), 1930 (162): 169–184, doi:10.1515/crll.1930.162.169, ISSN 0075-4102, S2CID 199546442{{citation}}: CS1 maint: unrecognized language (link)
• Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.