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An integer associated to the character of a representation of the Galois group of a finite extension of a local field. Let  $  K $
be a field that is complete with respect to a discrete valuation, with residue class field  $  k $
of characteristic  $  p \geq  0 $. 
Let  $  L/K $
be a Galois extension of degree  $  n $
with Galois group  $  G $
and suppose that the residue class field extension is separable. If  $  \chi $
is the character of some finite-dimensional complex representation of  $  G $, 
its conductor  $  f ( \chi ) $
is defined by the formula:

$$ 
f ( \chi )  = \ 
\sum _ {i = 0 } ^  \infty  

\frac{n _ {i} }{n _ {0} }

( \chi ( 1) - \chi ( G _ {i} )),
$$

where

$$ 
G _ {i}  = \ 
\{ {g \in G } : {
\nu _ {L} ( g ( x) - x) \geq  i + 1  \textrm{ for } \ 
\textrm{ all }  x \in L \ 
\textrm{ with } \ 
\nu _ {L} ( x) \geq  0 } \}
,
$$

$$ 
n _ {i}  =  | G _ {i} |,\  \chi ( G _ {i} )
 =  n _ {i}  ^ {-1} \sum _ {g \in G _ {i} } \chi ( g) ,
$$

where  $  \nu _ {L} $
is the corresponding valuation of  $  L $. 
If  $  p $
does not divide  $  n $, 
then  $  G _ {i} = \{ 1 \} $
for  $  i > 0 $
and  $  f ( \chi ) = \chi ( 1) - \chi ( G _ {0} ) $. 
If  $  \chi $
is the character of a rational representation  $  M $, 
then  $  \chi ( G _ {i} ) =  \mathop{\rm dim}  M ^ {G _ {i} } $. 
The conductor  $  f ( \chi ) $
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  $  \mathfrak p _ {k} ^ {f ( \chi ) } $, 
where  $  f ( \chi ) $
is the conductor of a character  $  \chi $
of the Galois group of an extension of local fields, is also called the Artin conductor of  $  \chi $. 
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  $  L $-
functions, cf. [[L-function| $  L $-
function]].

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