View source for Unramified ideal

<!--
u0957901.png
$#A+1 = 34 n = 0
$#C+1 = 34 : ~/encyclopedia/old_files/data/U095/U.0905790 Unramified ideal
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

{{MSC|11S15}}

A [[Prime ideal|prime ideal]]  $  \mathfrak P $
of an algebraic number field  $  K $(
cf. also [[Algebraic number|Algebraic number]]; [[Number field|Number field]]) lying over a prime number  $  p $
such that the principal ideal  $  ( p) $
has in  $  K $
a product decomposition into prime ideals of the form

$$ 
( p)  =  \mathfrak P _ {1} ^ {e _ {1} } \dots \mathfrak P _ {s} ^ {e _ {s} } ,
$$

where

$$ 
\mathfrak P _ {1}  =  \mathfrak P \  \textrm{ and } \ \ 
\mathfrak P _ {2} \dots \mathfrak P _ {s}  \neq  \mathfrak P ,
$$

and  $  e _ {1} = 1 $. 
More accurately, such an ideal is called absolutely unramified. In general, let  $  A $
be a [[Dedekind ring|Dedekind ring]] with [[field of fractions]]  $  k $, 
let  $  K $
be a finite extension of  $  k $
and let  $  B $
be the integral closure of  $  A $
in  $  K $(
cf. [[Integral extension of a ring|Integral extension of a ring]]). A prime ideal  $  \mathfrak P $
of  $  B $
lying over an ideal  $  \mathfrak Y $
of  $  A $
is unramified in the extension  $  K / k $
if

$$ 
\mathfrak Y B  =  \mathfrak P _ {1} ^ {e _ {1} } \dots \mathfrak P _ {s} ^ {e _ {s} } ,
$$

where  $  \mathfrak P _ {1} \dots \mathfrak P _ {s} $
are pairwise distinct prime ideals of  $  B $, 
$  \mathfrak P _ {1} = \mathfrak P $
and  $  e _ {1} = 1 $. 
If all ideals  $  \mathfrak P _ {1} \dots \mathfrak P _ {s} $
are unramified, then one occasionally says that  $  \mathfrak Y $
remains unramified in  $  K / k $. 
For a Galois extension  $  K / k $, 
an ideal  $  \mathfrak P $
of  $  B $
is unramified if and only if the decomposition group of  $  \mathfrak P $
in the Galois group  $  G ( K / k ) $
is the same as the Galois group of the extension of the residue class field  $  ( B/ \mathfrak P ) / ( A/ \mathfrak Y ) $. 
In any finite extension of algebraic number fields all ideals except finitely many are unramified.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Z.I. Borevich,   I.R. Shafarevich,   "Number theory" , Acad. Press  (1966)  (Translated from Russian)  (German translation: Birkhäuser, 1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lang,   "Algebraic number theory" , Addison-Wesley  (1970)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press  (1986)</TD></TR></table>