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A [[Prime ideal|prime ideal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u0957901.png" /> of an algebraic number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u0957902.png" /> (cf. also [[Algebraic number|Algebraic number]]; [[Number field|Number field]]) lying over a prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u0957903.png" /> such that the principal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u0957904.png" /> has in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u0957905.png" /> a product decomposition into prime ideals of the form
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<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/u/u095/u095790/u0957906.png" /></td> </tr></table>
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{{MSC|11S15}}
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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
 
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/u/u095/u095790/u0957907.png" /></td> </tr></table>
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$$
 +
\mathfrak P _ {1}  = \mathfrak P \  \textrm{ and } \ \
 +
\mathfrak P _ {2} \dots \mathfrak P _ {s}  \neq  \mathfrak P ,
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u0957908.png" />. More accurately, such an ideal is called absolutely unramified. In general, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u0957909.png" /> be a [[Dedekind ring|Dedekind ring]] with [[field of fractions]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579010.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579011.png" /> be a finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579012.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579013.png" /> be the integral closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579015.png" /> (cf. [[Integral extension of a ring|Integral extension of a ring]]). A prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579017.png" /> lying over an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579019.png" /> is unramified in the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579020.png" /> if
+
and $  e _ {1} = 1 $.  
 +
More accurately, such an ideal is called absolutely unramified. In general, let $  A $
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be a [[Dedekind ring|Dedekind ring]] with [[field of fractions]] $  k $,  
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let $  K $
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be a finite extension of $  k $
 +
and let $  B $
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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 $
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if
  
<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/u/u095/u095790/u09579021.png" /></td> </tr></table>
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$$
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\mathfrak Y B  = \mathfrak P _ {1} ^ {e _ {1} } \dots \mathfrak P _ {s} ^ {e _ {s} } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579022.png" /> are pairwise distinct prime ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579025.png" />. If all ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579026.png" /> are unramified, then one occasionally says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579027.png" /> remains unramified in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579028.png" />. For a Galois extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579029.png" />, an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579030.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579031.png" /> is unramified if and only if the decomposition group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579032.png" /> in the Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579033.png" /> is the same as the Galois group of the extension of the residue class field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579034.png" />. In any finite extension of algebraic number fields all ideals except finitely many are unramified.
+
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====
 
====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>
 
<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>

Latest revision as of 08:27, 6 June 2020


2020 Mathematics Subject Classification: Primary: 11S15 [MSN][ZBL]

A prime ideal $ \mathfrak P $ of an algebraic number field $ K $( cf. also Algebraic number; 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 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). 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

[1] Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966)
[2] S. Lang, "Algebraic number theory" , Addison-Wesley (1970)
[3] J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986)
How to Cite This Entry:
Unramified ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unramified_ideal&oldid=35063
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article