View source for Conductor of a character

An integer associated to the character of a representation of the Galois group of a finite extension of a local field. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c0245601.png" /> be a field that is complete with respect to a discrete valuation, with residue class field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c0245602.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c0245603.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c0245604.png" /> be a Galois extension of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c0245605.png" /> with Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c0245606.png" /> and suppose that the residue class field extension is separable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c0245607.png" /> is the character of some finite-dimensional complex representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c0245608.png" />, its conductor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c0245609.png" /> is defined by the formula:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456010.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456011.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456012.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456013.png" /> is the corresponding valuation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456015.png" /> does not divide <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456016.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456017.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456019.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456020.png" /> is the character of a rational representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456021.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456022.png" />. The conductor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456023.png" /> is a non-negative integer.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press  (1967)  pp. Chapt. VI</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Artin,   J. Tate,   "Class field theory" , Benjamin  (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.-P. Serre,   "Local fields" , Springer  (1979)  (Translated from French)</TD></TR></table>



====Comments====
The ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456025.png" /> is the conductor of a character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456026.png" /> of the Galois group of an extension of local fields, is also called the Artin conductor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456027.png" />. There is a corresponding notion for extensions of global fields obtained by taking a suitable product over all finite primes, cf. [[#References|[a1]]], p. 126. It plays an important role in the theory of Artin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456028.png" />-functions, cf. [[L-function|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456029.png" />-function]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Neukirch,   "Class field theory" , Springer  (1986)</TD></TR></table>