Namespaces
Variants
Actions

Difference between revisions of "Frobenius automorphism"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
An element of a Galois group of a special type. It plays a fundamental role in class field theory. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f0417601.png" /> is an algebraic extension of a finite field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f0417602.png" />. Then the Frobenius automorphism is the automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f0417603.png" /> defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f0417604.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f0417605.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f0417606.png" /> (the cardinality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f0417607.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f0417608.png" /> is a finite extension, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f0417609.png" /> generates the Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176010.png" />. For an infinite extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176012.png" /> is a topological generator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176015.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176016.png" />.
+
<!--
 +
f0417601.png
 +
$#A+1 = 63 n = 0
 +
$#C+1 = 63 : ~/encyclopedia/old_files/data/F041/F.0401760 Frobenius automorphism
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176017.png" /> is a local field with a finite residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176018.png" />, and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176019.png" /> is an unramified extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176020.png" />. Then the Frobenius automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176021.png" /> of the extension of residue fields can be uniquely lifted to an automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176022.png" />, called the Frobenius automorphism of the unramified extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176023.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176024.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176025.png" /> be the ring of integers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176026.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176027.png" /> be a maximal ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176028.png" />. Then the Frobenius automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176029.png" /> is uniquely determined by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176030.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176031.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176032.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176033.png" /> is an arbitrary Galois extension of local fields, then sometimes any automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176034.png" /> that induces a Frobenius automorphism in the sense indicated above on the maximal unramified subextension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176035.png" /> is called a Frobenius automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176036.png" />.
+
{{TEX|auto}}
 +
{{TEX|done}}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176037.png" /> be a Galois extension of global fields, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176038.png" /> be a prime ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176039.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176040.png" /> be some prime ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176041.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176042.png" />. Suppose also that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176043.png" /> is unramified in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176044.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176045.png" /> is the Frobenius automorphism of the unramified extension of local fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176046.png" />. If one identifies the Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176047.png" /> with the decomposition subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176048.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176049.png" />, one can regard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176050.png" /> as an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176051.png" />. This element is called the Frobenius automorphism corresponding to the prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176052.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176053.png" /> is a finite extension, then, according to the Chebotarev density theorem, for any automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176054.png" /> there is an infinite number of prime ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176055.png" />, unramified in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176056.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176057.png" />. For an Abelian extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176058.png" />, the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176059.png" /> depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176060.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176061.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176062.png" /> and is called the Artin symbol of the prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041760/f04176063.png" />.
+
An element of a Galois group of a special type. It plays a fundamental role in class field theory. Suppose that  $  L $
 +
is an algebraic extension of a finite field  $  K $.  
 +
Then the Frobenius automorphism is the automorphism  $  \phi _ {L/K} $
 +
defined by the formula  $  \phi _ {L/K} ( a) = a  ^ {q} $
 +
for all  $  a \in L $,
 +
where  $  q = | K | $(
 +
the cardinality of  $  K $).  
 +
If  $  L/K $
 +
is a finite extension, then  $  \phi _ {L/K} $
 +
generates the Galois group  $  \mathop{\rm Gal} ( L/K) $.  
 +
For an infinite extension  $  L/K $,
 +
$  \phi _ {L/K} $
 +
is a topological generator of  $  \mathop{\rm Gal} ( L/K) $.
 +
If  $  L \supset E \supset K $
 +
and  $  [ E:  K] < \infty $,
 +
then  $  \phi _ {L/E} = \phi _ {L/K} ^ {[ E: K] } $.
 +
 
 +
Suppose that  $  k $
 +
is a local field with a finite residue field  $  \overline{k}\; $,
 +
and that  $  K $
 +
is an unramified extension of $  k $.  
 +
Then the Frobenius automorphism  $  \phi _ {\overline{K}\; / \overline{k}\; }  $
 +
of the extension of residue fields can be uniquely lifted to an automorphism  $  \phi _ {K/k} \in  \mathop{\rm Gal} ( K/k) $,
 +
called the Frobenius automorphism of the unramified extension  $  K/k $.  
 +
Let  $  | \overline{k}\; | = q $,
 +
let  $  {\mathcal O} _ {K} $
 +
be the ring of integers of  $  K $,  
 +
and let $  \mathfrak p $
 +
be a maximal ideal in  $  {\mathcal O} _ {K} $.
 +
Then the Frobenius automorphism  $  \phi _ {K/k} $
 +
is uniquely determined by the condition  $  \phi _ {K/k} ( a) \equiv a  ^ {q} $
 +
$  \mathop{\rm mod}  \mathfrak p $
 +
for every  $  a \in {\mathcal O} _ {k} $.  
 +
If  $  K/k $
 +
is an arbitrary Galois extension of local fields, then sometimes any automorphism  $  \phi \in  \mathop{\rm Gal} ( K/k) $
 +
that induces a Frobenius automorphism in the sense indicated above on the maximal unramified subextension of  $  K $
 +
is called a Frobenius automorphism of  $  K/k $.
 +
 
 +
Let  $  K/k $
 +
be a Galois extension of global fields, let  $  \mathfrak p $
 +
be a prime ideal of  $  k $,
 +
and let  $  \mathfrak P $
 +
be some prime ideal of $  K $
 +
over $  \mathfrak p $.  
 +
Suppose also that $  \mathfrak P $
 +
is unramified in $  K/k $
 +
and that $  \phi _ {\mathfrak P} \in  \mathop{\rm Gal} ( K _ {\mathfrak P} /k _ {\mathfrak p} ) $
 +
is the Frobenius automorphism of the unramified extension of local fields $  K _ {\mathfrak P} /k _ {\mathfrak p} $.  
 +
If one identifies the Galois group $  \mathop{\rm Gal} ( K _ {\mathfrak P} /k _ {\mathfrak p} ) $
 +
with the decomposition subgroup of $  \mathfrak P $
 +
in $  \mathop{\rm Gal} ( K/k) $,  
 +
one can regard $  \phi _ {\mathfrak P} $
 +
as an element of $  \mathop{\rm Gal} ( K/k) $.  
 +
This element is called the Frobenius automorphism corresponding to the prime ideal $  \mathfrak P $.  
 +
If $  K/k $
 +
is a finite extension, then, according to the Chebotarev density theorem, for any automorphism $  \sigma \in  \mathop{\rm Gal} ( K/k) $
 +
there is an infinite number of prime ideals $  \mathfrak P $,  
 +
unramified in $  K/k $,  
 +
such that $  \sigma = \phi _ {\mathfrak P} $.  
 +
For an Abelian extension $  K/k $,  
 +
the element $  \phi _ {\mathfrak P} $
 +
depends only on $  \mathfrak p $.  
 +
In this case $  \phi _ {\mathfrak P} $
 +
is denoted by $  ( \mathfrak p , K/k) $
 +
and is called the Artin symbol of the prime ideal $  \mathfrak p $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Weil,  "Basic number theory" , Springer  (1974)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Weil,  "Basic number theory" , Springer  (1974)</TD></TR></table>

Revision as of 19:40, 5 June 2020


An element of a Galois group of a special type. It plays a fundamental role in class field theory. Suppose that $ L $ is an algebraic extension of a finite field $ K $. Then the Frobenius automorphism is the automorphism $ \phi _ {L/K} $ defined by the formula $ \phi _ {L/K} ( a) = a ^ {q} $ for all $ a \in L $, where $ q = | K | $( the cardinality of $ K $). If $ L/K $ is a finite extension, then $ \phi _ {L/K} $ generates the Galois group $ \mathop{\rm Gal} ( L/K) $. For an infinite extension $ L/K $, $ \phi _ {L/K} $ is a topological generator of $ \mathop{\rm Gal} ( L/K) $. If $ L \supset E \supset K $ and $ [ E: K] < \infty $, then $ \phi _ {L/E} = \phi _ {L/K} ^ {[ E: K] } $.

Suppose that $ k $ is a local field with a finite residue field $ \overline{k}\; $, and that $ K $ is an unramified extension of $ k $. Then the Frobenius automorphism $ \phi _ {\overline{K}\; / \overline{k}\; } $ of the extension of residue fields can be uniquely lifted to an automorphism $ \phi _ {K/k} \in \mathop{\rm Gal} ( K/k) $, called the Frobenius automorphism of the unramified extension $ K/k $. Let $ | \overline{k}\; | = q $, let $ {\mathcal O} _ {K} $ be the ring of integers of $ K $, and let $ \mathfrak p $ be a maximal ideal in $ {\mathcal O} _ {K} $. Then the Frobenius automorphism $ \phi _ {K/k} $ is uniquely determined by the condition $ \phi _ {K/k} ( a) \equiv a ^ {q} $ $ \mathop{\rm mod} \mathfrak p $ for every $ a \in {\mathcal O} _ {k} $. If $ K/k $ is an arbitrary Galois extension of local fields, then sometimes any automorphism $ \phi \in \mathop{\rm Gal} ( K/k) $ that induces a Frobenius automorphism in the sense indicated above on the maximal unramified subextension of $ K $ is called a Frobenius automorphism of $ K/k $.

Let $ K/k $ be a Galois extension of global fields, let $ \mathfrak p $ be a prime ideal of $ k $, and let $ \mathfrak P $ be some prime ideal of $ K $ over $ \mathfrak p $. Suppose also that $ \mathfrak P $ is unramified in $ K/k $ and that $ \phi _ {\mathfrak P} \in \mathop{\rm Gal} ( K _ {\mathfrak P} /k _ {\mathfrak p} ) $ is the Frobenius automorphism of the unramified extension of local fields $ K _ {\mathfrak P} /k _ {\mathfrak p} $. If one identifies the Galois group $ \mathop{\rm Gal} ( K _ {\mathfrak P} /k _ {\mathfrak p} ) $ with the decomposition subgroup of $ \mathfrak P $ in $ \mathop{\rm Gal} ( K/k) $, one can regard $ \phi _ {\mathfrak P} $ as an element of $ \mathop{\rm Gal} ( K/k) $. This element is called the Frobenius automorphism corresponding to the prime ideal $ \mathfrak P $. If $ K/k $ is a finite extension, then, according to the Chebotarev density theorem, for any automorphism $ \sigma \in \mathop{\rm Gal} ( K/k) $ there is an infinite number of prime ideals $ \mathfrak P $, unramified in $ K/k $, such that $ \sigma = \phi _ {\mathfrak P} $. For an Abelian extension $ K/k $, the element $ \phi _ {\mathfrak P} $ depends only on $ \mathfrak p $. In this case $ \phi _ {\mathfrak P} $ is denoted by $ ( \mathfrak p , K/k) $ and is called the Artin symbol of the prime ideal $ \mathfrak p $.

References

[1] A. Weil, "Basic number theory" , Springer (1974)
How to Cite This Entry:
Frobenius automorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frobenius_automorphism&oldid=46992
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