conductor-discriminant formula

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

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

where $\backslash mathfrak\{f\}(\backslash chi)$ equals the global Artin conductor of $\backslash chi$.{{sfn|Neukirch|1999|loc=VII.11.9}}

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

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

• {{Citation | last1=Artin | first1=Emil | author1-link=Emil Artin | title=Die gruppentheoretische Struktur der Diskriminanten algebraischer Zahlkörper. | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002171422 | language=German | doi=10.1515/crll.1931.164.1 | zbl= 0001.00801 | year=1931 | journal=Journal für die Reine und Angewandte Mathematik | issn=0075-4102 | volume=1931 | issue=164 | pages=1–11| s2cid=117731518 | url-access=subscription }}
• {{Citation | last1=Hasse | first1=H. | author1-link=Helmut Hasse | title=Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. I: Klassenkörpertheorie. | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002127768 | language=German | year=1926 | journal=Jahresbericht der Deutschen Mathematiker-Vereinigung | volume=35 | pages=1–55}}
• {{Citation | last1=Hasse | first1=H. | author1-link=Helmut Hasse | title=Führer, Diskriminante und Verzweigungskörper relativ-Abelscher Zahlkörper. | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002171198 | language=German | doi=10.1515/crll.1930.162.169 | year=1930 | journal=Journal für die reine und angewandte Mathematik | issn=0075-4102 | volume=1930 | issue=162 | pages=169–184| s2cid=199546442 | url-access=subscription }}
• {{Neukirch ANT}}

Category:Algebraic number theory