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

Statement

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

[math]\displaystyle{ \mathfrak{d}_{L/K} = \prod_{\chi \in \mathrm{Irr}(G)}\mathfrak{f}(\chi)^{\chi(1)}, }[/math]

where [math]\displaystyle{ \mathfrak{f}(\chi) }[/math] equals the global Artin conductor of [math]\displaystyle{ \chi }[/math].

Example

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

[math]\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}. }[/math]

References

• Artin, Emil (1931), "Die gruppentheoretische Struktur der Diskriminanten algebraischer Zahlkörper." (in German), Journal für die Reine und Angewandte Mathematik 1931 (164): 1–11, doi:10.1515/crll.1931.164.1, ISSN 0075-4102, http://resolver.sub.uni-goettingen.de/purl?GDZPPN002171422 
• Hasse, H. (1926), "Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. I: Klassenkörpertheorie." (in German), Jahresbericht der Deutschen Mathematiker-Vereinigung 35: 1–55, http://resolver.sub.uni-goettingen.de/purl?GDZPPN002127768 
• Hasse, H. (1930), "Führer, Diskriminante und Verzweigungskörper relativ-Abelscher Zahlkörper." (in German), Journal für die reine und angewandte Mathematik 1930 (162): 169–184, doi:10.1515/crll.1930.162.169, ISSN 0075-4102, http://resolver.sub.uni-goettingen.de/purl?GDZPPN002171198 
• Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8.