Namespaces
Variants
Actions

Difference between revisions of "Place of a field"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p0727501.png" /> with values in a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p0727502.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p0727504.png" />-valued place of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p0727505.png" />''
+
<!--
 +
p0727501.png
 +
$#A+1 = 37 n = 0
 +
$#C+1 = 37 : ~/encyclopedia/old_files/data/P072/P.0702750 Place of a field
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p0727506.png" /> satisfying the conditions
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/p/p072/p072750/p0727507.png" /></td> </tr></table>
+
'' $  K $
 +
with values in a field  $  L $,
 +
$  L $-
 +
valued place of a field  $  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/p/p072/p072750/p0727508.png" /></td> </tr></table>
+
A mapping  $  f: K \rightarrow L \cup \{ \infty \} $
 +
satisfying the conditions
  
<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/p/p072/p072750/p0727509.png" /></td> </tr></table>
+
$$
 +
f ( 1)  = 1,
 +
$$
 +
 
 +
$$
 +
f ( a + b)  = f ( a) + f ( b),
 +
$$
 +
 
 +
$$
 +
f ( ab)  = f ( a) \cdot f ( b)
 +
$$
  
 
(provided that the expressions on the right-hand sides are defined). The following conventions are made:
 
(provided that the expressions on the right-hand sides are defined). The following conventions are made:
  
<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/p/p072/p072750/p07275010.png" /></td> </tr></table>
+
$$
 +
\infty \cdot \infty  = \infty ,
 +
$$
  
<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/p/p072/p072750/p07275011.png" /></td> </tr></table>
+
$$
 +
c + \infty  = \infty + c  = \infty ,\  c \in L,
 +
$$
  
<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/p/p072/p072750/p07275012.png" /></td> </tr></table>
+
$$
 +
c \cdot \infty  = \infty \cdot c  = \infty ,\  c \in L,\  c \neq 0,
 +
$$
  
while the expressions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275015.png" /> are undefined.
+
while the expressions $  \infty + \infty $,  
 +
0 \cdot \infty $
 +
and $  \infty \cdot 0 $
 +
are undefined.
  
An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275017.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275018.png" /> is called finite in the place <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275019.png" />; the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275020.png" /> of finite elements is a subring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275021.png" />, and the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275022.png" /> is a ring homomorphism. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275023.png" /> is a [[Local ring|local ring]], its maximal ideal is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275024.png" />.
+
An element $  a $
 +
in $  K $
 +
for which $  f ( a) \in L $
 +
is called finite in the place $  f $;  
 +
the set $  A $
 +
of finite elements is a subring of $  K $,  
 +
and the mapping $  f: A \rightarrow L $
 +
is a ring homomorphism. The ring $  A $
 +
is a [[Local ring|local ring]], its maximal ideal is $  \mathfrak m = \{ {a \in K } : {f ( a) = 0 } \} $.
  
A place <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275025.png" /> determines a [[Valuation|valuation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275026.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275027.png" /> with group of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275028.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275030.png" /> are, respectively, the groups of invertible elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275032.png" />). The ring of this valuation is the same as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275033.png" />. Conversely, any valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275034.png" /> of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275035.png" /> determines a place of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275036.png" /> with values in the residue class field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275037.png" />. Here, the ring of finite elements is the same as the ring of (integers of) the valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072750/p07275038.png" />.
+
A place $  f $
 +
determines a [[Valuation|valuation]] $  v $
 +
of $  K $
 +
with group of values $  K  ^ {*} /A  ^ {*} $(
 +
where $  K  ^ {*} = K \setminus  \{ 0 \} $
 +
and $  A  ^ {*} = A \setminus  \mathfrak m $
 +
are, respectively, the groups of invertible elements of $  K $
 +
and $  A $).  
 +
The ring of this valuation is the same as $  A $.  
 +
Conversely, any valuation $  v $
 +
of a field $  K $
 +
determines a place of $  K $
 +
with values in the residue class field of $  v $.  
 +
Here, the ring of finite elements is the same as the ring of (integers of) the valuation $  v $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1984)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1984)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Algebra" , '''2''' , Wiley  (1977)  pp. Chapt. 9</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Algebra" , '''2''' , Wiley  (1977)  pp. Chapt. 9</TD></TR></table>

Latest revision as of 08:06, 6 June 2020


$ K $ with values in a field $ L $, $ L $- valued place of a field $ K $

A mapping $ f: K \rightarrow L \cup \{ \infty \} $ satisfying the conditions

$$ f ( 1) = 1, $$

$$ f ( a + b) = f ( a) + f ( b), $$

$$ f ( ab) = f ( a) \cdot f ( b) $$

(provided that the expressions on the right-hand sides are defined). The following conventions are made:

$$ \infty \cdot \infty = \infty , $$

$$ c + \infty = \infty + c = \infty ,\ c \in L, $$

$$ c \cdot \infty = \infty \cdot c = \infty ,\ c \in L,\ c \neq 0, $$

while the expressions $ \infty + \infty $, $ 0 \cdot \infty $ and $ \infty \cdot 0 $ are undefined.

An element $ a $ in $ K $ for which $ f ( a) \in L $ is called finite in the place $ f $; the set $ A $ of finite elements is a subring of $ K $, and the mapping $ f: A \rightarrow L $ is a ring homomorphism. The ring $ A $ is a local ring, its maximal ideal is $ \mathfrak m = \{ {a \in K } : {f ( a) = 0 } \} $.

A place $ f $ determines a valuation $ v $ of $ K $ with group of values $ K ^ {*} /A ^ {*} $( where $ K ^ {*} = K \setminus \{ 0 \} $ and $ A ^ {*} = A \setminus \mathfrak m $ are, respectively, the groups of invertible elements of $ K $ and $ A $). The ring of this valuation is the same as $ A $. Conversely, any valuation $ v $ of a field $ K $ determines a place of $ K $ with values in the residue class field of $ v $. Here, the ring of finite elements is the same as the ring of (integers of) the valuation $ v $.

References

[1] S. Lang, "Algebra" , Addison-Wesley (1984)

Comments

References

[a1] P.M. Cohn, "Algebra" , 2 , Wiley (1977) pp. Chapt. 9
How to Cite This Entry:
Place of a field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Place_of_a_field&oldid=48183
This article was adapted from an original article by Yu.G. Zarkhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article