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A [[Prime ideal|prime ideal]] in a [[Dedekind ring|Dedekind ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r0772301.png" /> which divides the discriminant of a finite [[Separable extension|separable extension]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r0772302.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r0772303.png" /> is the field of fractions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r0772304.png" />. Such ideals are the only ideals that are ramified in the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r0772305.png" />. A prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r0772306.png" /> of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r0772307.png" /> is ramified in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r0772308.png" /> if the following product representation holds in the integral closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r0772309.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723010.png" /> in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723011.png" />:
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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/r/r077/r077230/r07723012.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723013.png" /> are prime ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723014.png" /> and at least one of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723015.png" /> is greater than 1. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723016.png" /> is called the ramification index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723017.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723018.png" />.
+
A [[Prime ideal|prime ideal]] in a [[Dedekind ring|Dedekind ring]]  $  A $
 +
which divides the discriminant of a finite [[Separable extension|separable extension]]  $  K/k $,
 +
where $  k $
 +
is the field of fractions of  $  A $.  
 +
Such ideals are the only ideals that are ramified in the extension  $  K/k $.  
 +
A prime ideal  $  \mathfrak p $
 +
of a ring  $  A $
 +
is ramified in  $  K/k $
 +
if the following product representation holds in the integral closure  $  B $
 +
of $  A $
 +
in the field  $  K $:
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723019.png" /> is a [[Galois extension|Galois extension]] with Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723022.png" /> is precisely the order of the inertia subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723023.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723024.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723025.png" />:
+
$$
 +
\mathfrak p B  = \
 +
\mathfrak P _ {1} ^ {e _ {1} } \dots
 +
\mathfrak P _ {s} ^ {e _ {s} } ,
 +
$$
  
<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/r/r077/r077230/r07723026.png" /></td> </tr></table>
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where  $  \mathfrak P _ {1} \dots \mathfrak P _ {s} $
 +
are prime ideals in  $  B $
 +
and at least one of the numbers  $  e _ {i} $
 +
is greater than 1. The number  $  e _ {i} $
 +
is called the ramification index of  $  \mathfrak P _ {i} $
 +
over  $  \mathfrak p $.
  
Other, more refined, characteristics of the ramification are given by the higher ramification groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723028.png" /> defined as follows:
+
If  $  K/k $
 +
is a [[Galois extension|Galois extension]] with Galois group  $  G ( K/k) $,  
 +
then  $  e _ {1} = \dots = e _ {s} $
 +
and  $  e _ {i} $
 +
is precisely the order of the inertia subgroup  $  T ( \mathfrak P _ {i} ) $
 +
of  $  \mathfrak P _ {i} $
 +
in  $  G ( K/k) $:
  
<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/r/r077/r077230/r07723029.png" /></td> </tr></table>
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$$
 +
T ( \mathfrak P _ {i} )  = \
 +
\{ {\sigma \in G ( K/k) } : {
 +
\sigma a - a \in \mathfrak P _ {i} \
 +
\textrm{ for }  \textrm{ all } \
 +
a \in B } \}
 +
.
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723030.png" />; by Minkowski's theorem, for any finite extension of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723031.png" /> of rational numbers there exists a ramified prime ideal. This is not true for arbitrary algebraic number fields: If the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723032.png" /> has class number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723033.png" />, i.e. has a non-trivial ideal class group, then there exist unramified extensions over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723034.png" />, i.e. extensions having no ramified prime ideal. An example of such an extension is the Hilbert class field of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723035.png" />; e.g., the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723036.png" /> is the Hilbert class field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723037.png" /> and is unramified over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077230/r07723038.png" />.
+
Other, more refined, characteristics of the ramification are given by the higher ramification groups  $  T ( \mathfrak P _ {i} ) _ {n} \subset  T ( \mathfrak P _ {i} ) $,
 +
$  n = 1, 2 \dots $
 +
defined as follows:
 +
 
 +
$$
 +
T ( \mathfrak P _ {i} ) _ {n}  = \
 +
\{ {\sigma \in G ( K/k) } : {
 +
\sigma a - a \in \mathfrak P _ {i} ^ {n + 1 } \
 +
\textrm{ for }  \textrm{ all }  a \in B } \}
 +
.
 +
$$
 +
 
 +
Let  $  A = \mathbf Z $;  
 +
by Minkowski's theorem, for any finite extension of the field $  \mathbf Q $
 +
of rational numbers there exists a ramified prime ideal. This is not true for arbitrary algebraic number fields: If the field $  k $
 +
has class number $  h > 1 $,  
 +
i.e. has a non-trivial ideal class group, then there exist unramified extensions over $  k $,  
 +
i.e. extensions having no ramified prime ideal. An example of such an extension is the Hilbert class field of the field $  k $;  
 +
e.g., the field $  \mathbf Q ( \sqrt 5 , \sqrt - 5 ) $
 +
is the Hilbert class field of $  \mathbf Q ( \sqrt - 5 ) $
 +
and is unramified over $  \mathbf Q ( \sqrt - 5 ) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Z.I. Borevich,  I.R. Shafarevich,  "Number theory" , Acad. Press  (1987)  (Translated from Russian)  (German translation: Birkhäuser, 1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press  (1986)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Lang,  "Algebraic number theory" , Addison-Wesley  (1970)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Z.I. Borevich,  I.R. Shafarevich,  "Number theory" , Acad. Press  (1987)  (Translated from Russian)  (German translation: Birkhäuser, 1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press  (1986)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Lang,  "Algebraic number theory" , Addison-Wesley  (1970)</TD></TR></table>

Latest revision as of 08:09, 6 June 2020


A prime ideal in a Dedekind ring $ A $ which divides the discriminant of a finite separable extension $ K/k $, where $ k $ is the field of fractions of $ A $. Such ideals are the only ideals that are ramified in the extension $ K/k $. A prime ideal $ \mathfrak p $ of a ring $ A $ is ramified in $ K/k $ if the following product representation holds in the integral closure $ B $ of $ A $ in the field $ K $:

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

where $ \mathfrak P _ {1} \dots \mathfrak P _ {s} $ are prime ideals in $ B $ and at least one of the numbers $ e _ {i} $ is greater than 1. The number $ e _ {i} $ is called the ramification index of $ \mathfrak P _ {i} $ over $ \mathfrak p $.

If $ K/k $ is a Galois extension with Galois group $ G ( K/k) $, then $ e _ {1} = \dots = e _ {s} $ and $ e _ {i} $ is precisely the order of the inertia subgroup $ T ( \mathfrak P _ {i} ) $ of $ \mathfrak P _ {i} $ in $ G ( K/k) $:

$$ T ( \mathfrak P _ {i} ) = \ \{ {\sigma \in G ( K/k) } : { \sigma a - a \in \mathfrak P _ {i} \ \textrm{ for } \textrm{ all } \ a \in B } \} . $$

Other, more refined, characteristics of the ramification are given by the higher ramification groups $ T ( \mathfrak P _ {i} ) _ {n} \subset T ( \mathfrak P _ {i} ) $, $ n = 1, 2 \dots $ defined as follows:

$$ T ( \mathfrak P _ {i} ) _ {n} = \ \{ {\sigma \in G ( K/k) } : { \sigma a - a \in \mathfrak P _ {i} ^ {n + 1 } \ \textrm{ for } \textrm{ all } a \in B } \} . $$

Let $ A = \mathbf Z $; by Minkowski's theorem, for any finite extension of the field $ \mathbf Q $ of rational numbers there exists a ramified prime ideal. This is not true for arbitrary algebraic number fields: If the field $ k $ has class number $ h > 1 $, i.e. has a non-trivial ideal class group, then there exist unramified extensions over $ k $, i.e. extensions having no ramified prime ideal. An example of such an extension is the Hilbert class field of the field $ k $; e.g., the field $ \mathbf Q ( \sqrt 5 , \sqrt - 5 ) $ is the Hilbert class field of $ \mathbf Q ( \sqrt - 5 ) $ and is unramified over $ \mathbf Q ( \sqrt - 5 ) $.

References

[1] Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1987) (Translated from Russian) (German translation: Birkhäuser, 1966)
[2] J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986)
[3] S. Lang, "Algebraic number theory" , Addison-Wesley (1970)
How to Cite This Entry:
Ramified prime ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ramified_prime_ideal&oldid=48419
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