Namespaces
Variants
Actions

Difference between revisions of "Inertial prime number"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (link)
m (tex encoded by computer)
 
Line 1: Line 1:
''inert prime number, in an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508002.png" />''
+
<!--
 +
i0508002.png
 +
$#A+1 = 48 n = 0
 +
$#C+1 = 48 : ~/encyclopedia/old_files/data/I050/I.0500800 Inertial prime number,
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
A prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508003.png" /> such that the principal ideal generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508004.png" /> remains prime in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508005.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508006.png" /> is a finite extension of the field of rational numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508007.png" />; in other words, the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508008.png" /> is prime in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508009.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080010.png" /> is the ring of integers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080011.png" />. In this case one also says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080012.png" /> is inert in the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080013.png" />. By analogy, a prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080014.png" /> of a Dedekind ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080015.png" /> is said to be inert in the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080017.png" /> is the [[field of fractions]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080019.png" /> is a finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080020.png" />, if the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080022.png" /> is the integral closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080023.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080024.png" />, is prime.
+
{{TEX|auto}}
 +
{{TEX|done}}
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080025.png" /> is a Galois extension with Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080026.png" />, then for any ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080027.png" /> of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080028.png" />, a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080029.png" /> of the decomposition group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080030.png" /> of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080031.png" /> is defined which is called the inertia group (see [[Ramified prime ideal|Ramified prime ideal]]). The extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080032.png" /> is a maximal intermediate extension in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080033.png" /> in which the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080034.png" /> is inert.
+
''inert prime number, in an extension  $  K / \mathbf Q $''
 +
 
 +
A prime number  $  p $
 +
such that the principal ideal generated by  $  p $
 +
remains prime in  $  K / \mathbf Q $,
 +
where  $  K $
 +
is a finite extension of the field of rational numbers  $  \mathbf Q $;
 +
in other words, the ideal  $  ( p) $
 +
is prime in  $  B $,
 +
where  $  B $
 +
is the ring of integers of  $  K $.  
 +
In this case one also says that  $  p $
 +
is inert in the extension  $  K / \mathbf Q $.  
 +
By analogy, a prime ideal  $  \mathfrak p $
 +
of a Dedekind ring  $  A $
 +
is said to be inert in the extension  $  K / k $,
 +
where  $  k $
 +
is the [[field of fractions]] of  $  A $
 +
and  $  K $
 +
is a finite extension of  $  k $,
 +
if the ideal  $  \mathfrak p B $,
 +
where  $  B $
 +
is the integral closure of  $  A $
 +
in  $  K $,
 +
is prime.
 +
 
 +
If  $  K / k $
 +
is a Galois extension with Galois group $  G $,  
 +
then for any ideal $  \mathfrak P $
 +
of the ring $  B \subset  K $,  
 +
a subgroup $  T _ {\mathfrak P }  $
 +
of the decomposition group $  G _ {\mathfrak P }  $
 +
of the ideal $  \mathfrak P $
 +
is defined which is called the inertia group (see [[Ramified prime ideal|Ramified prime ideal]]). The extension $  K ^ {T _ {\mathfrak P }  } / K ^ {G _ {\mathfrak P }  } $
 +
is a maximal intermediate extension in $  K / k $
 +
in which the ideal $  \mathfrak P \cap K ^ {G _ {\mathfrak P }  } $
 +
is inert.
  
 
In cyclic extensions of algebraic number fields there always exist infinitely many inert prime ideals.
 
In cyclic extensions of algebraic number fields there always exist infinitely many inert prime ideals.
Line 9: Line 53:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,  "Algebraic number theory" , Addison-Wesley  (1970)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Weyl,  "Algebraic theory of numbers" , Princeton Univ. Press  (1959)</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">  S. Lang,  "Algebraic number theory" , Addison-Wesley  (1970)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Weyl,  "Algebraic theory of numbers" , Princeton Univ. Press  (1959)</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>
 
 
  
 
====Comments====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080035.png" /> be a Galois extension with Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080036.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080037.png" /> be a prime ideal of (the ring of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080038.png" />) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080039.png" />. The decomposition group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080040.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080041.png" />. The subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080042.png" /> is the inertia group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080043.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080044.png" />. It is a normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080045.png" />. The subfields of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080046.png" /> which, according to Galois theory, correspond to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080048.png" />, are called respectively the decomposition field and inertia field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080049.png" />.
+
Let $  K / k $
 +
be a Galois extension with Galois group $  G $.  
 +
Let $  \mathfrak P $
 +
be a prime ideal of (the ring of integers $  A _ {K} $)  
 +
of $  K $.  
 +
The decomposition group of $  \mathfrak P $
 +
is defined by $  G _ {\mathfrak P }  = \{ {\sigma \in G } : {\mathfrak P  ^  \sigma  = \mathfrak P } \} $.  
 +
The subgroup $  I _ {\mathfrak P }  = \{ {\sigma \in G _ {\mathfrak P }  } : {a  ^  \sigma  \equiv a  \mathop{\rm mod}  \mathfrak P  \textrm{ for  all  }  a \in B } \} $
 +
is the inertia group of $  \mathfrak P $
 +
over $  k $.  
 +
It is a normal subgroup of $  G _ {\mathfrak P }  $.  
 +
The subfields of $  K $
 +
which, according to Galois theory, correspond to $  G _ {\mathfrak P }  $
 +
and $  I _ {\mathfrak P }  $,  
 +
are called respectively the decomposition field and inertia field of $  \mathfrak P $.

Latest revision as of 22:12, 5 June 2020


inert prime number, in an extension $ K / \mathbf Q $

A prime number $ p $ such that the principal ideal generated by $ p $ remains prime in $ K / \mathbf Q $, where $ K $ is a finite extension of the field of rational numbers $ \mathbf Q $; in other words, the ideal $ ( p) $ is prime in $ B $, where $ B $ is the ring of integers of $ K $. In this case one also says that $ p $ is inert in the extension $ K / \mathbf Q $. By analogy, a prime ideal $ \mathfrak p $ of a Dedekind ring $ A $ is said to be inert in the extension $ K / k $, where $ k $ is the field of fractions of $ A $ and $ K $ is a finite extension of $ k $, if the ideal $ \mathfrak p B $, where $ B $ is the integral closure of $ A $ in $ K $, is prime.

If $ K / k $ is a Galois extension with Galois group $ G $, then for any ideal $ \mathfrak P $ of the ring $ B \subset K $, a subgroup $ T _ {\mathfrak P } $ of the decomposition group $ G _ {\mathfrak P } $ of the ideal $ \mathfrak P $ is defined which is called the inertia group (see Ramified prime ideal). The extension $ K ^ {T _ {\mathfrak P } } / K ^ {G _ {\mathfrak P } } $ is a maximal intermediate extension in $ K / k $ in which the ideal $ \mathfrak P \cap K ^ {G _ {\mathfrak P } } $ is inert.

In cyclic extensions of algebraic number fields there always exist infinitely many inert prime ideals.

References

[1] S. Lang, "Algebraic number theory" , Addison-Wesley (1970)
[2] H. Weyl, "Algebraic theory of numbers" , Princeton Univ. Press (1959)
[3] J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986)

Comments

Let $ K / k $ be a Galois extension with Galois group $ G $. Let $ \mathfrak P $ be a prime ideal of (the ring of integers $ A _ {K} $) of $ K $. The decomposition group of $ \mathfrak P $ is defined by $ G _ {\mathfrak P } = \{ {\sigma \in G } : {\mathfrak P ^ \sigma = \mathfrak P } \} $. The subgroup $ I _ {\mathfrak P } = \{ {\sigma \in G _ {\mathfrak P } } : {a ^ \sigma \equiv a \mathop{\rm mod} \mathfrak P \textrm{ for all } a \in B } \} $ is the inertia group of $ \mathfrak P $ over $ k $. It is a normal subgroup of $ G _ {\mathfrak P } $. The subfields of $ K $ which, according to Galois theory, correspond to $ G _ {\mathfrak P } $ and $ I _ {\mathfrak P } $, are called respectively the decomposition field and inertia field of $ \mathfrak P $.

How to Cite This Entry:
Inertial prime number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inertial_prime_number&oldid=35051
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