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 $\mathrm {Irr} (G)$ 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 $U_{\mathbf {Q} _{p}}^{(n_{p})}\subseteq N_{L_{\mathfrak {p}}^{\chi }/\mathbf {Q} _{p}}(U_{L_{\mathfrak {p}}^{\chi }})$. If $p>2$, the Galois group $G(L_{\mathfrak {p}}/\mathbf {Q} _{p})=G(L/\mathbf {Q} )=(\mathbf {Z} /p^{n})^{\times }$ is cyclic of order $\varphi (p^{n})$, and by local class field theory and using that $U_{\mathbf {Q} _{p}}/U_{\mathbf {Q} _{p}}^{(k)}=(\mathbf {Z} /p^{k})^{\times }$ 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 $\varphi (p^{i})-\varphi (p^{i-1})$ 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

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

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
• 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
• 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
• 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.