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'' $  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|local ring]], its maximal ideal is  $  \mathfrak m = \{ {a \in K } : {f ( a) = 0 } \} $.

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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,   "Algebra" , Addison-Wesley  (1984)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,   "Algebra" , '''2''' , Wiley  (1977)  pp. Chapt. 9</TD></TR></table>