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''logarithmic norm, norm on a field''
 
''logarithmic norm, norm on a field''
  
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v0960101.png" /> from a [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v0960102.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v0960103.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v0960104.png" /> is a totally ordered Abelian group, the adjoined element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v0960105.png" /> is assumed to be larger than any element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v0960106.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v0960107.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v0960108.png" />. Here the valuation must satisfy the following conditions:
+
A mapping $  v : K \rightarrow \Gamma _  \infty  $
 +
from a [[Field|field]] $  K $
 +
into $  \Gamma _  \infty  = \Gamma \cup \{ \infty \} $,  
 +
where $  \Gamma $
 +
is a totally ordered Abelian group, the adjoined element $  \infty $
 +
is assumed to be larger than any element of $  \Gamma $,  
 +
and $  x + \infty = \infty + x = \infty $
 +
for all $  x \in \Gamma $.  
 +
Here the valuation must satisfy the following conditions:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v0960109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601010.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601011.png" />;
+
1) $  v ( 0) = \infty $,
 +
v ( x) < \infty $
 +
for $  x \neq 0 $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601012.png" />;
+
2) $  v ( a \cdot b ) = v ( a) + v ( b) $;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601013.png" />.
+
3) v ( a - b ) \geq  \min ( v ( a) , v ( b) ) $.
  
The image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601014.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601015.png" /> is a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601016.png" />, called the value group of the valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601017.png" />. Throughout what follows it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601018.png" />.
+
The image of $  K  ^ {*} = K \setminus  \{ 0 \} $
 +
under v $
 +
is a subgroup of $  \Gamma $,  
 +
called the value group of the valuation v $.  
 +
Throughout what follows it is assumed that $  v ( K  ^ {*} ) = \Gamma $.
  
By the same axioms one defines logarithmic valuations of rings. Every ring with a non-Archimedean norm (cf. [[Norm on a field|Norm on a field]]) can be made into a logarithmically-valued ring if one passes in the groupoid of values from the multiplicative to the additive notation and reverses the order. The element 0 is then naturally denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601019.png" />. The reverse transition from a ring with a logarithmic valuation to one with a non-Archimedean norm is also possible. If in a ring a non-Archimedean [[Real norm|real norm]] is given, then the corresponding transition can be obtained by replacing each positive real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601020.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601021.png" />. The resulting logarithmic valuation is also called real.
+
By the same axioms one defines logarithmic valuations of rings. Every ring with a non-Archimedean norm (cf. [[Norm on a field|Norm on a field]]) can be made into a logarithmically-valued ring if one passes in the groupoid of values from the multiplicative to the additive notation and reverses the order. The element 0 is then naturally denoted by the symbol $  \infty $.  
 +
The reverse transition from a ring with a logarithmic valuation to one with a non-Archimedean norm is also possible. If in a ring a non-Archimedean [[Real norm|real norm]] is given, then the corresponding transition can be obtained by replacing each positive real number $  \alpha $
 +
by $  -  \mathop{\rm log}  \alpha $.  
 +
The resulting logarithmic valuation is also called real.
  
Two valuations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601023.png" /> are said to be equivalent if there is an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601024.png" /> of ordered groups such that for all non-zero elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601025.png" />,
+
Two valuations $  v _ {1} : K \rightarrow \Gamma _  \infty  ^ {1} $
 +
and  $  v _ {2} : K \rightarrow \Gamma _  \infty  ^ {2} $
 +
are said to be equivalent if there is an isomorphism $  \phi : \Gamma  ^ {1} \rightarrow \Gamma  ^ {2} $
 +
of ordered groups such that for all non-zero elements $  x \in 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/v/v096/v096010/v09601026.png" /></td> </tr></table>
+
$$
 +
v _ {2} ( x)  = \phi ( v _ {1} ( x) ) .
 +
$$
  
The set of those elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601028.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601029.png" /> is a subring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601030.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601031.png" />, called the valuation ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601032.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601033.png" />. It is always a [[Local ring|local ring]]. The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601034.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601035.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601036.png" /> form a maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601037.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601038.png" />, called the valuation ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601039.png" />. The quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601040.png" />, which is a field, is called the residue field of the valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601041.png" />.
+
The set of those elements $  x $
 +
of $  K $
 +
for which $  v ( x) \geq  0 $
 +
is a subring $  A $
 +
of $  K $,  
 +
called the valuation ring of v $
 +
in $  K $.  
 +
It is always a [[Local ring|local ring]]. The elements $  x $
 +
of $  K $
 +
for which v ( x) > 0 $
 +
form a maximal ideal $  m _ {v} $
 +
of $  A $,  
 +
called the valuation ideal of v $.  
 +
The quotient ring $  A / m _ {v} $,  
 +
which is a field, is called the residue field of the valuation v $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601043.png" /> be two valuations on a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601044.png" />. The rings of these valuations, regarded as subrings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601045.png" />, are the same if and only if these valuations are equivalent. Thus, knowing all valuations of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601046.png" /> (up to equivalence) is the same as knowing all subrings that occur as valuation rings for this field. A subring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601047.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601048.png" /> is a valuation ring for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601049.png" /> if and only if for every non-zero element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601050.png" /> at least one of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601052.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601053.png" />. Thus, a valuation ring can be defined abstractly as an integral ring ([[Integral domain|integral domain]]) that satisfies this condition relative to its field of fractions. Every such ring is the ring of the so-called canonical valuation for its field of fractions, for which the value group is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601055.png" /> is the multiplicative group of invertible elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601056.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601057.png" /> is ordered by divisibility.
+
Let v _ {1} $
 +
and v _ {2} $
 +
be two valuations on a field $  K $.  
 +
The rings of these valuations, regarded as subrings of $  K $,  
 +
are the same if and only if these valuations are equivalent. Thus, knowing all valuations of a field $  K $(
 +
up to equivalence) is the same as knowing all subrings that occur as valuation rings for this field. A subring $  A $
 +
of $  K $
 +
is a valuation ring for $  K $
 +
if and only if for every non-zero element $  x \in K $
 +
at least one of $  x $
 +
and $  x  ^ {-} 1 $
 +
belongs to $  A $.  
 +
Thus, a valuation ring can be defined abstractly as an integral ring ([[Integral domain|integral domain]]) that satisfies this condition relative to its field of fractions. Every such ring is the ring of the so-called canonical valuation for its field of fractions, for which the value group is $  K  ^ {*} / U $,  
 +
where $  U $
 +
is the multiplicative group of invertible elements of $  A $,  
 +
and $  K  ^ {*} / U $
 +
is ordered by divisibility.
  
A valuation ring can be defined in yet another way. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601058.png" /> are two local rings with maximal ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601060.png" />, respectively, then one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601061.png" /> dominates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601062.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601063.png" />. Dominance is a partial order relation on the set of subrings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601064.png" />. The maximal elements of this set are exactly the valuation rings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601065.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601066.png" /> is a valuation ring and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601067.png" /> is a ring with the same field of fractions as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601068.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601069.png" /> is also a valuation ring and is the localization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601070.png" /> with respect to some prime ideal.
+
A valuation ring can be defined in yet another way. If $  A \subseteq B \subseteq K $
 +
are two local rings with maximal ideals $  m $
 +
and $  n $,  
 +
respectively, then one says that $  B $
 +
dominates $  A $
 +
if $  m \subset  n $.  
 +
Dominance is a partial order relation on the set of subrings of $  K $.  
 +
The maximal elements of this set are exactly the valuation rings of $  K $.  
 +
If $  A $
 +
is a valuation ring and $  B \supset A $
 +
is a ring with the same field of fractions as $  A $,  
 +
then $  B $
 +
is also a valuation ring and is the localization of $  A $
 +
with respect to some prime ideal.
  
 
==Examples of valuations.==
 
==Examples of valuations.==
 
  
 
1) The valuation defined by
 
1) The valuation defined by
  
<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/v/v096/v096010/v09601071.png" /></td> </tr></table>
+
$$
 +
v ( x)  = \
 +
\left \{
  
 
is called improper or trivial. Any valuation of a finite field is trivial.
 
is called improper or trivial. Any valuation of a finite field is trivial.
  
2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601072.png" /> be a field and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601073.png" /> be the field of Laurent series over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601074.png" />. Associating to a series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601075.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601076.png" />, its order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601077.png" /> (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601078.png" /> to the null series) is a valuation with value group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601079.png" /> (the additive group of integers) and valuation ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601080.png" />.
+
2) Let $  k $
 +
be a field and let $  K = k ( ( t ) ) $
 +
be the field of Laurent series over $  k $.  
 +
Associating to a series $  a _ {n} t  ^ {n} + a _ {n+} 1 t  ^ {n+} 1 + \dots $,  
 +
where $  a _ {n} \neq 0 $,  
 +
its order $  n $(
 +
and $  \infty $
 +
to the null series) is a valuation with value group $  \mathbf Z $(
 +
the additive group of integers) and valuation ring $  k [ [ t ] ] $.
  
A valuation with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601081.png" /> is called discrete; about their valuation rings see [[Discretely-normed ring|Discretely-normed ring]]. For a description of all valuations of the field of rational numbers, see [[#References|[4]]].
+
A valuation with values in $  \mathbf Z $
 +
is called discrete; about their valuation rings see [[Discretely-normed ring|Discretely-normed ring]]. For a description of all valuations of the field of rational numbers, see [[#References|[4]]].
  
For each totally ordered Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601082.png" /> there is a valuation of a certain field with value group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601083.png" />.
+
For each totally ordered Abelian group $  \Gamma $
 +
there is a valuation of a certain field with value group $  \Gamma $.
  
 
==Ideals in valuation rings.==
 
==Ideals in valuation rings.==
 
The set of ideals in a valuation ring is totally ordered by inclusion; every ideal of finite type is principal, that is, a valuation ring is a [[Bezout ring|Bezout ring]]. A more complete description of the structure of ideals in a valuation ring can be given in terms of the value group of the valuation.
 
The set of ideals in a valuation ring is totally ordered by inclusion; every ideal of finite type is principal, that is, a valuation ring is a [[Bezout ring|Bezout ring]]. A more complete description of the structure of ideals in a valuation ring can be given in terms of the value group of the valuation.
  
A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601084.png" /> of a totally ordered set is said to be major if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601086.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601087.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601088.png" /> be the ring of a valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601089.png" /> of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601090.png" /> with value group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601091.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601092.png" /> be the sub-semi-group of positive elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601093.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601094.png" /> be a major set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601095.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601096.png" /> is a bijection between the set of major subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601097.png" /> and the set of ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601098.png" />. Principal ideals correspond to majors having a minimal element. Prime ideals also correspond to majors of special form, namely: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v09601099.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010100.png" /> is the positive part of a [[Convex subgroup|convex subgroup]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010101.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010102.png" />. Thus, there is a one-to-one correspondence between the prime ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010103.png" /> and the convex subgroups of the value group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010104.png" />.
+
A subset $  M $
 +
of a totally ordered set is said to be major if $  x \in M $
 +
and $  y > x $
 +
imply $  y \in M $.  
 +
Let $  A $
 +
be the ring of a valuation v $
 +
of a field $  k $
 +
with value group $  \Gamma $,  
 +
let $  \Gamma  ^ {+} $
 +
be the sub-semi-group of positive elements of $  \Gamma $,  
 +
and let $  M $
 +
be a major set in $  \Gamma  ^ {+} $.  
 +
The mapping $  M \mapsto \alpha ( M) = \{ {x } : {x \in K,  v ( x) ; M \cup \{ \infty \} } \} $
 +
is a bijection between the set of major subsets of $  \Gamma  ^ {+} $
 +
and the set of ideals of $  A $.  
 +
Principal ideals correspond to majors having a minimal element. Prime ideals also correspond to majors of special form, namely: $  M _ {H} = \Gamma  ^ {+} \setminus  H  ^ {+} $,  
 +
where $  H  ^ {+} $
 +
is the positive part of a [[Convex subgroup|convex subgroup]] $  H $
 +
of $  \Gamma $.  
 +
Thus, there is a one-to-one correspondence between the prime ideals of $  A $
 +
and the convex subgroups of the value group $  \Gamma $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010105.png" /> be the prime ideal corresponding to a convex subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010106.png" />. Then the composite mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010107.png" /> is a valuation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010108.png" /> with valuation ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010109.png" /> and valuation ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010110.png" />; moreover, on the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010111.png" /> there is an induced valuation with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010112.png" /> and valuation ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010113.png" />. In this manner a valuation splits into simpler ones. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010114.png" /> be a valuation ring. Then the prime spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010115.png" /> without the zero (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010116.png" />) is a totally ordered set, and its type is called the height or rank of the corresponding valuation. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010117.png" /> is finite, then the height of the valuation is the number of elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010118.png" />, and this is the same as the number of proper convex subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010119.png" />. A valuation of finite rank can be reduced to valuations of rank 1. The latter are characterized by the fact that their value groups are Archimedean (cf. [[Archimedean group|Archimedean group]]), that is, they are isomorphic to a subgroup of the additive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010120.png" /> of real numbers. In this case the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010121.png" /> is an [[ultrametric]] norm on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010122.png" />.
+
Let $  p $
 +
be the prime ideal corresponding to a convex subgroup $  H $.  
 +
Then the composite mapping $  K \rightarrow  ^ {v} \Gamma \rightarrow \Gamma / H $
 +
is a valuation of $  K $
 +
with valuation ring $  A _ {p} $
 +
and valuation ideal $  p A _ {p} $;  
 +
moreover, on the field $  A _ {p} / p A _ {p} $
 +
there is an induced valuation with values in $  H $
 +
and valuation ring $  A / p $.  
 +
In this manner a valuation splits into simpler ones. Let $  A $
 +
be a valuation ring. Then the prime spectrum of $  A $
 +
without the zero ( $  \mathop{\rm Spec}  A \setminus  ( 0) $)
 +
is a totally ordered set, and its type is called the height or rank of the corresponding valuation. If $  \mathop{\rm Spec}  A $
 +
is finite, then the height of the valuation is the number of elements in $  \mathop{\rm Spec}  A \setminus  ( 0) $,  
 +
and this is the same as the number of proper convex subgroups of $  \Gamma $.  
 +
A valuation of finite rank can be reduced to valuations of rank 1. The latter are characterized by the fact that their value groups are Archimedean (cf. [[Archimedean group|Archimedean group]]), that is, they are isomorphic to a subgroup of the additive group $  \mathbf R $
 +
of real numbers. In this case the mapping $  x \mapsto  \mathop{\rm exp} ( - v ( x) ) $
 +
is an [[ultrametric]] norm on $  K $.
  
An important property of valuation rings is that they are integrally closed. Moreover, for an arbitrary integral ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010123.png" /> its integral closure is equal to the intersection of all valuation rings containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010124.png" />. A valuation ring is totally integrally closed if and only if its valuation is real, that is, has rank 1. A valuation ring is Noetherian if and only if the valuation is discrete.
+
An important property of valuation rings is that they are integrally closed. Moreover, for an arbitrary integral ring $  A $
 +
its integral closure is equal to the intersection of all valuation rings containing $  A $.  
 +
A valuation ring is totally integrally closed if and only if its valuation is real, that is, has rank 1. A valuation ring is Noetherian if and only if the valuation is discrete.
  
 
==Valuations and topologies.==
 
==Valuations and topologies.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010125.png" /> be a valuation on a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010126.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010127.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010128.png" />. The collection of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010129.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010130.png" />, is a fundamental system of neighbourhoods of zero for a topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010131.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010132.png" />, which is said to be the topology determined by the valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010133.png" />. It is separable and disconnected. The topology induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010134.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010135.png" /> is, as a rule, different from that of a local ring. For a non-trivial valuation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010136.png" /> the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010137.png" /> is locally compact if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010138.png" /> is discrete, the valuation ring is complete, and the residue field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010139.png" /> is finite; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010140.png" /> is then compact. The completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010141.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010142.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010143.png" /> is a field; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010144.png" /> can be extended by continuity to a valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010145.png" />, and the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010146.png" /> is the same as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010147.png" />. The valuation ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010148.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010149.png" /> is the completion of the valuation ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010150.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010151.png" />.
+
Let $  v : K \rightarrow \Gamma _  \infty  $
 +
be a valuation on a field $  K $
 +
and let $  V _  \gamma  = \{ {x } : {x \in K,  v ( x) > \gamma } \} $,  
 +
where $  \gamma \in \Gamma $.  
 +
The collection of all $  V _  \gamma  $,  
 +
$  \gamma \in \Gamma $,  
 +
is a fundamental system of neighbourhoods of zero for a topology $  \tau _ {v} $
 +
of $  K $,  
 +
which is said to be the topology determined by the valuation v $.  
 +
It is separable and disconnected. The topology induced by $  \tau _ {v} $
 +
on $  A $
 +
is, as a rule, different from that of a local ring. For a non-trivial valuation of $  K $
 +
the topology $  \tau _ {v} $
 +
is locally compact if and only if v $
 +
is discrete, the valuation ring is complete, and the residue field of v $
 +
is finite; $  A $
 +
is then compact. The completion $  \widehat{K}  $
 +
of $  K $
 +
relative to $  \tau _ {v} $
 +
is a field; v $
 +
can be extended by continuity to a valuation $  \widehat{v}  : \widehat{K}  \rightarrow \Gamma _  \infty  $,  
 +
and the topology of $  \widehat{K}  $
 +
is the same as $  \tau _ {\widehat{v}  }  $.  
 +
The valuation ring $  \widehat{A}  $
 +
of $  \widehat{v}  $
 +
is the completion of the valuation ring $  A $
 +
of v $.
  
Two valuations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010152.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010153.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010154.png" /> are called independent if the topologies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010155.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010156.png" /> are distinct; this is equivalent to the fact that the valuation rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010157.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010158.png" /> generate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010159.png" />. Inequivalent valuations of height 1 are always independent. There is an approximation theorem for valuations: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010160.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010161.png" />, be independent valuations, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010162.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010163.png" />. Then there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010164.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010165.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010166.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010167.png" />.
+
Two valuations v _ {1} $
 +
and v _ {2} $
 +
of $  K $
 +
are called independent if the topologies $  \tau _ {v _ {1}  } $
 +
and $  \tau _ {v _ {2}  } $
 +
are distinct; this is equivalent to the fact that the valuation rings $  A _ {v _ {1}  } $
 +
and $  A _ {v _ {2}  } $
 +
generate $  K $.  
 +
Inequivalent valuations of height 1 are always independent. There is an approximation theorem for valuations: Let $  v _ {i} : K \rightarrow \Gamma _  \infty  ^ {i} $,  
 +
$  1 \leq  i \leq  n $,  
 +
be independent valuations, let $  a _ {i} \in K $,  
 +
$  \alpha _ {i} \in \Gamma  ^ {i} $.  
 +
Then there is an element $  x $
 +
in $  K $
 +
such that v _ {i} ( x - a _ {i} ) \geq  \alpha _ {i} $
 +
for all $  i $.
  
 
==Extension of valuations.==
 
==Extension of valuations.==
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010168.png" /> is a valuation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010169.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010170.png" /> is a subfield of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010171.png" />, then the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010172.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010173.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010174.png" /> is a valuation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010175.png" />, and its value group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010176.png" /> is a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010177.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010178.png" /> is then called an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010179.png" />. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010180.png" /> is a valuation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010181.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010182.png" /> is an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010183.png" />, then there is always a valuation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010184.png" /> that extends <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010185.png" />. The index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010186.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010187.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010188.png" /> is called the ramification index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010189.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010190.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010191.png" />. The residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010192.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010193.png" /> can be identified with a subfield of the residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010194.png" /> and the degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010195.png" /> of the extension is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010196.png" /> and is called the residue degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010197.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010198.png" />. An extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010199.png" /> of a valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010200.png" /> is said to be immediate if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010201.png" />.
+
If $  v : L \rightarrow \Gamma _  \infty  ^  \prime  $
 +
is a valuation of $  L $
 +
and $  K $
 +
is a subfield of $  L $,  
 +
then the restriction $  v = v ^  \prime  \mid  _ {K} $
 +
of v ^  \prime  $
 +
to $  K $
 +
is a valuation of $  K $,  
 +
and its value group $  \Gamma $
 +
is a subgroup of $  \Gamma  ^  \prime  $;  
 +
v ^  \prime  $
 +
is then called an extension of v $.  
 +
Conversely, if v $
 +
is a valuation on $  K $
 +
and $  L $
 +
is an extension of $  K $,  
 +
then there is always a valuation of $  L $
 +
that extends v $.  
 +
The index $  [ \Gamma  ^  \prime  : \Gamma ] $
 +
of $  \Gamma $
 +
in $  \Gamma  ^  \prime  $
 +
is called the ramification index of v ^  \prime  $
 +
with respect to v $
 +
and is denoted by $  e ( v  ^  \prime  / v ) $.  
 +
The residue field $  k _ {v} $
 +
of v $
 +
can be identified with a subfield of the residue field $  k _ {v ^  \prime  } $
 +
and the degree $  [ k _ {v  ^  \prime  } : k _ {v} ] $
 +
of the extension is denoted by $  f ( v  ^  \prime  / v ) $
 +
and is called the residue degree of v ^  \prime  $
 +
relative to v $.  
 +
An extension v ^  \prime  $
 +
of a valuation v $
 +
is said to be immediate if $  e ( v  ^  \prime  / v ) = f ( v  ^  \prime  / v ) = 1 $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010202.png" /> be an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010203.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010204.png" /> be the set of all extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010205.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010206.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010207.png" /> is a finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010208.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010209.png" />, then the set of all extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010210.png" /> is finite, and
+
Let $  L $
 +
be an extension of $  K $
 +
and let $  \{ {v _ {i} } : {i \in I } \} $
 +
be the set of all extensions of v $
 +
to $  L $.  
 +
If $  L $
 +
is a finite extension of $  K $
 +
of degree $  n $,  
 +
then the set of all extensions of v $
 +
is finite, and
  
<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/v/v096/v096010/v096010211.png" /></td> </tr></table>
+
$$
 +
\sum _ {i \in I }
 +
e ( v _ {i} / v )
 +
f ( v _ {i} / v )  \leq  n .
 +
$$
  
In several cases equality holds, for example when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010212.png" /> is discrete and either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010213.png" /> is complete or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010214.png" /> is separable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010215.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010216.png" /> is a normal extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010217.png" />, then the extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010218.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010219.png" /> are permuted transitively by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010220.png" />-automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010221.png" />; in particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010222.png" /> is a [[purely inseparable extension]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010223.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010224.png" /> has only one extension. In the case of an arbitrary extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010225.png" /> and an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010226.png" /> of a valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010227.png" />, the transcendence degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010228.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010229.png" /> is greater than or equal to the sum
+
In several cases equality holds, for example when v $
 +
is discrete and either $  K $
 +
is complete or $  L $
 +
is separable over $  K $.  
 +
If $  L $
 +
is a normal extension of $  K $,  
 +
then the extensions of v $
 +
to $  L $
 +
are permuted transitively by the $  K $-
 +
automorphisms of $  L $;  
 +
in particular, if $  L $
 +
is a [[purely inseparable extension]] of $  K $,  
 +
then v $
 +
has only one extension. In the case of an arbitrary extension $  K \subset  L $
 +
and an extension v ^  \prime  $
 +
of a valuation v $,  
 +
the transcendence degree of $  L $
 +
over $  K $
 +
is greater than or equal to the sum
  
<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/v/v096/v096010/v096010230.png" /></td> </tr></table>
+
$$
 +
S ( {v  ^  \prime  } / v ) + r ( {v  ^  \prime  } / v ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010231.png" /> is the transcendence degree of the extension of the residue field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010232.png" /> over that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010233.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010234.png" /> is the dimension of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096010/v096010235.png" />.
+
where $  S ( v  ^  \prime  / v ) $
 +
is the transcendence degree of the extension of the residue field of v ^  \prime  $
 +
over that of v $
 +
and $  r ( v  ^  \prime  / v ) $
 +
is the dimension of the space $  ( \Gamma _ {v  ^  \prime  } / \Gamma _ {v} ) \otimes \mathbf Q $.
  
 
The concept of a valuation was introduced and studied by W. Krull in [[#References|[1]]]. It is also widely used in algebraic geometry. Thus, in terms of "valuation rings" one can construct the abstract Riemann surface of a field (cf. [[#References|[3]]]).
 
The concept of a valuation was introduced and studied by W. Krull in [[#References|[1]]]. It is also widely used in algebraic geometry. Thus, in terms of "valuation rings" one can construct the abstract Riemann surface of a field (cf. [[#References|[3]]]).
Line 69: Line 307:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> W. Krull, "Allgemeine Bewertungstheorie" ''J. Reine Angew. Math.'' , '''167''' (1932) pp. 160–196 {{MR|}} {{ZBL|0004.09802}} {{ZBL|58.0148.02}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) {{MR|0360549}} {{ZBL|0279.13001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''2''' , Springer (1975) {{MR|0389876}} {{MR|0384768}} {{ZBL|0313.13001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) {{MR|0158000}} {{ZBL|0121.25901}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> W. Krull, "Allgemeine Bewertungstheorie" ''J. Reine Angew. Math.'' , '''167''' (1932) pp. 160–196 {{MR|}} {{ZBL|0004.09802}} {{ZBL|58.0148.02}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) {{MR|0360549}} {{ZBL|0279.13001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''2''' , Springer (1975) {{MR|0389876}} {{MR|0384768}} {{ZBL|0313.13001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) {{MR|0158000}} {{ZBL|0121.25901}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> O. Endler, "Valuation theory" , Springer (1972) {{MR|0357379}} {{ZBL|0257.12111}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> O. Endler, "Valuation theory" , Springer (1972) {{MR|0357379}} {{ZBL|0257.12111}} </TD></TR></table>

Latest revision as of 08:27, 6 June 2020


logarithmic norm, norm on a field

A mapping $ v : K \rightarrow \Gamma _ \infty $ from a field $ K $ into $ \Gamma _ \infty = \Gamma \cup \{ \infty \} $, where $ \Gamma $ is a totally ordered Abelian group, the adjoined element $ \infty $ is assumed to be larger than any element of $ \Gamma $, and $ x + \infty = \infty + x = \infty $ for all $ x \in \Gamma $. Here the valuation must satisfy the following conditions:

1) $ v ( 0) = \infty $, $ v ( x) < \infty $ for $ x \neq 0 $;

2) $ v ( a \cdot b ) = v ( a) + v ( b) $;

3) $ v ( a - b ) \geq \min ( v ( a) , v ( b) ) $.

The image of $ K ^ {*} = K \setminus \{ 0 \} $ under $ v $ is a subgroup of $ \Gamma $, called the value group of the valuation $ v $. Throughout what follows it is assumed that $ v ( K ^ {*} ) = \Gamma $.

By the same axioms one defines logarithmic valuations of rings. Every ring with a non-Archimedean norm (cf. Norm on a field) can be made into a logarithmically-valued ring if one passes in the groupoid of values from the multiplicative to the additive notation and reverses the order. The element 0 is then naturally denoted by the symbol $ \infty $. The reverse transition from a ring with a logarithmic valuation to one with a non-Archimedean norm is also possible. If in a ring a non-Archimedean real norm is given, then the corresponding transition can be obtained by replacing each positive real number $ \alpha $ by $ - \mathop{\rm log} \alpha $. The resulting logarithmic valuation is also called real.

Two valuations $ v _ {1} : K \rightarrow \Gamma _ \infty ^ {1} $ and $ v _ {2} : K \rightarrow \Gamma _ \infty ^ {2} $ are said to be equivalent if there is an isomorphism $ \phi : \Gamma ^ {1} \rightarrow \Gamma ^ {2} $ of ordered groups such that for all non-zero elements $ x \in K $,

$$ v _ {2} ( x) = \phi ( v _ {1} ( x) ) . $$

The set of those elements $ x $ of $ K $ for which $ v ( x) \geq 0 $ is a subring $ A $ of $ K $, called the valuation ring of $ v $ in $ K $. It is always a local ring. The elements $ x $ of $ K $ for which $ v ( x) > 0 $ form a maximal ideal $ m _ {v} $ of $ A $, called the valuation ideal of $ v $. The quotient ring $ A / m _ {v} $, which is a field, is called the residue field of the valuation $ v $.

Let $ v _ {1} $ and $ v _ {2} $ be two valuations on a field $ K $. The rings of these valuations, regarded as subrings of $ K $, are the same if and only if these valuations are equivalent. Thus, knowing all valuations of a field $ K $( up to equivalence) is the same as knowing all subrings that occur as valuation rings for this field. A subring $ A $ of $ K $ is a valuation ring for $ K $ if and only if for every non-zero element $ x \in K $ at least one of $ x $ and $ x ^ {-} 1 $ belongs to $ A $. Thus, a valuation ring can be defined abstractly as an integral ring (integral domain) that satisfies this condition relative to its field of fractions. Every such ring is the ring of the so-called canonical valuation for its field of fractions, for which the value group is $ K ^ {*} / U $, where $ U $ is the multiplicative group of invertible elements of $ A $, and $ K ^ {*} / U $ is ordered by divisibility.

A valuation ring can be defined in yet another way. If $ A \subseteq B \subseteq K $ are two local rings with maximal ideals $ m $ and $ n $, respectively, then one says that $ B $ dominates $ A $ if $ m \subset n $. Dominance is a partial order relation on the set of subrings of $ K $. The maximal elements of this set are exactly the valuation rings of $ K $. If $ A $ is a valuation ring and $ B \supset A $ is a ring with the same field of fractions as $ A $, then $ B $ is also a valuation ring and is the localization of $ A $ with respect to some prime ideal.

Examples of valuations.

1) The valuation defined by

$$ v ( x) = \ \left \{ is called improper or trivial. Any valuation of a finite field is trivial. 2) Let $ k $ be a field and let $ K = k ( ( t ) ) $ be the field of Laurent series over $ k $. Associating to a series $ a _ {n} t ^ {n} + a _ {n+} 1 t ^ {n+} 1 + \dots $, where $ a _ {n} \neq 0 $, its order $ n $( and $ \infty $ to the null series) is a valuation with value group $ \mathbf Z $( the additive group of integers) and valuation ring $ k [ [ t ] ] $. A valuation with values in $ \mathbf Z $ is called discrete; about their valuation rings see [[Discretely-normed ring|Discretely-normed ring]]. For a description of all valuations of the field of rational numbers, see [[#References|[4]]]. For each totally ordered Abelian group $ \Gamma $ there is a valuation of a certain field with value group $ \Gamma $. =='"`UNIQ--h-1--QINU`"'Ideals in valuation rings.== The set of ideals in a valuation ring is totally ordered by inclusion; every ideal of finite type is principal, that is, a valuation ring is a [[Bezout ring|Bezout ring]]. A more complete description of the structure of ideals in a valuation ring can be given in terms of the value group of the valuation. A subset $ M $ of a totally ordered set is said to be major if $ x \in M $ and $ y > x $ imply $ y \in M $. Let $ A $ be the ring of a valuation $ v $ of a field $ k $ with value group $ \Gamma $, let $ \Gamma ^ {+} $ be the sub-semi-group of positive elements of $ \Gamma $, and let $ M $ be a major set in $ \Gamma ^ {+} $. The mapping $ M \mapsto \alpha ( M) = \{ {x } : {x \in K, v ( x) ; M \cup \{ \infty \} } \} $ is a bijection between the set of major subsets of $ \Gamma ^ {+} $ and the set of ideals of $ A $. Principal ideals correspond to majors having a minimal element. Prime ideals also correspond to majors of special form, namely: $ M _ {H} = \Gamma ^ {+} \setminus H ^ {+} $, where $ H ^ {+} $ is the positive part of a [[Convex subgroup|convex subgroup]] $ H $ of $ \Gamma $. Thus, there is a one-to-one correspondence between the prime ideals of $ A $ and the convex subgroups of the value group $ \Gamma $. Let $ p $ be the prime ideal corresponding to a convex subgroup $ H $. Then the composite mapping $ K \rightarrow ^ {v} \Gamma \rightarrow \Gamma / H $ is a valuation of $ K $ with valuation ring $ A _ {p} $ and valuation ideal $ p A _ {p} $; moreover, on the field $ A _ {p} / p A _ {p} $ there is an induced valuation with values in $ H $ and valuation ring $ A / p $. In this manner a valuation splits into simpler ones. Let $ A $ be a valuation ring. Then the prime spectrum of $ A $ without the zero ( $ \mathop{\rm Spec} A \setminus ( 0) $) is a totally ordered set, and its type is called the height or rank of the corresponding valuation. If $ \mathop{\rm Spec} A $ is finite, then the height of the valuation is the number of elements in $ \mathop{\rm Spec} A \setminus ( 0) $, and this is the same as the number of proper convex subgroups of $ \Gamma $. A valuation of finite rank can be reduced to valuations of rank 1. The latter are characterized by the fact that their value groups are Archimedean (cf. [[Archimedean group|Archimedean group]]), that is, they are isomorphic to a subgroup of the additive group $ \mathbf R $ of real numbers. In this case the mapping $ x \mapsto \mathop{\rm exp} ( - v ( x) ) $ is an [[ultrametric]] norm on $ K $. An important property of valuation rings is that they are integrally closed. Moreover, for an arbitrary integral ring $ A $ its integral closure is equal to the intersection of all valuation rings containing $ A $. A valuation ring is totally integrally closed if and only if its valuation is real, that is, has rank 1. A valuation ring is Noetherian if and only if the valuation is discrete. =='"`UNIQ--h-2--QINU`"'Valuations and topologies.== Let $ v : K \rightarrow \Gamma _ \infty $ be a valuation on a field $ K $ and let $ V _ \gamma = \{ {x } : {x \in K, v ( x) > \gamma } \} $, where $ \gamma \in \Gamma $. The collection of all $ V _ \gamma $, $ \gamma \in \Gamma $, is a fundamental system of neighbourhoods of zero for a topology $ \tau _ {v} $ of $ K $, which is said to be the topology determined by the valuation $ v $. It is separable and disconnected. The topology induced by $ \tau _ {v} $ on $ A $ is, as a rule, different from that of a local ring. For a non-trivial valuation of $ K $ the topology $ \tau _ {v} $ is locally compact if and only if $ v $ is discrete, the valuation ring is complete, and the residue field of $ v $ is finite; $ A $ is then compact. The completion $ \widehat{K} $ of $ K $ relative to $ \tau _ {v} $ is a field; $ v $ can be extended by continuity to a valuation $ \widehat{v} : \widehat{K} \rightarrow \Gamma _ \infty $, and the topology of $ \widehat{K} $ is the same as $ \tau _ {\widehat{v} } $. The valuation ring $ \widehat{A} $ of $ \widehat{v} $ is the completion of the valuation ring $ A $ of $ v $. Two valuations $ v _ {1} $ and $ v _ {2} $ of $ K $ are called independent if the topologies $ \tau _ {v _ {1} } $ and $ \tau _ {v _ {2} } $ are distinct; this is equivalent to the fact that the valuation rings $ A _ {v _ {1} } $ and $ A _ {v _ {2} } $ generate $ K $. Inequivalent valuations of height 1 are always independent. There is an approximation theorem for valuations: Let $ v _ {i} : K \rightarrow \Gamma _ \infty ^ {i} $, $ 1 \leq i \leq n $, be independent valuations, let $ a _ {i} \in K $, $ \alpha _ {i} \in \Gamma ^ {i} $. Then there is an element $ x $ in $ K $ such that $ v _ {i} ( x - a _ {i} ) \geq \alpha _ {i} $ for all $ i $. =='"`UNIQ--h-3--QINU`"'Extension of valuations.== If $ v : L \rightarrow \Gamma _ \infty ^ \prime $ is a valuation of $ L $ and $ K $ is a subfield of $ L $, then the restriction $ v = v ^ \prime \mid _ {K} $ of $ v ^ \prime $ to $ K $ is a valuation of $ K $, and its value group $ \Gamma $ is a subgroup of $ \Gamma ^ \prime $; $ v ^ \prime $ is then called an extension of $ v $. Conversely, if $ v $ is a valuation on $ K $ and $ L $ is an extension of $ K $, then there is always a valuation of $ L $ that extends $ v $. The index $ [ \Gamma ^ \prime : \Gamma ] $ of $ \Gamma $ in $ \Gamma ^ \prime $ is called the ramification index of $ v ^ \prime $ with respect to $ v $ and is denoted by $ e ( v ^ \prime / v ) $. The residue field $ k _ {v} $ of $ v $ can be identified with a subfield of the residue field $ k _ {v ^ \prime } $ and the degree $ [ k _ {v ^ \prime } : k _ {v} ] $ of the extension is denoted by $ f ( v ^ \prime / v ) $ and is called the residue degree of $ v ^ \prime $ relative to $ v $. An extension $ v ^ \prime $ of a valuation $ v $ is said to be immediate if $ e ( v ^ \prime / v ) = f ( v ^ \prime / v ) = 1 $. Let $ L $ be an extension of $ K $ and let $ \{ {v _ {i} } : {i \in I } \} $ be the set of all extensions of $ v $ to $ L $. If $ L $ is a finite extension of $ K $ of degree $ n $, then the set of all extensions of $ v $ is finite, and $$ \sum _ {i \in I } e ( v _ {i} / v ) f ( v _ {i} / v ) \leq n . $$ In several cases equality holds, for example when $ v $ is discrete and either $ K $ is complete or $ L $ is separable over $ K $. If $ L $ is a normal extension of $ K $, then the extensions of $ v $ to $ L $ are permuted transitively by the $ K $- automorphisms of $ L $; in particular, if $ L $ is a [[purely inseparable extension]] of $ K $, then $ v $ has only one extension. In the case of an arbitrary extension $ K \subset L $ and an extension $ v ^ \prime $ of a valuation $ v $, the transcendence degree of $ L $ over $ K $ is greater than or equal to the sum $$ S ( {v ^ \prime } / v ) + r ( {v ^ \prime } / v ) , $$

where $ S ( v ^ \prime / v ) $ is the transcendence degree of the extension of the residue field of $ v ^ \prime $ over that of $ v $ and $ r ( v ^ \prime / v ) $ is the dimension of the space $ ( \Gamma _ {v ^ \prime } / \Gamma _ {v} ) \otimes \mathbf Q $.

The concept of a valuation was introduced and studied by W. Krull in [1]. It is also widely used in algebraic geometry. Thus, in terms of "valuation rings" one can construct the abstract Riemann surface of a field (cf. [3]).

References

[1] W. Krull, "Allgemeine Bewertungstheorie" J. Reine Angew. Math. , 167 (1932) pp. 160–196 Zbl 0004.09802 Zbl 58.0148.02
[2] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) MR0360549 Zbl 0279.13001
[3] O. Zariski, P. Samuel, "Commutative algebra" , 2 , Springer (1975) MR0389876 MR0384768 Zbl 0313.13001
[4] A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) MR0158000 Zbl 0121.25901

Comments

References

[a1] O. Endler, "Valuation theory" , Springer (1972) MR0357379 Zbl 0257.12111
How to Cite This Entry:
Valuation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Valuation&oldid=49103
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article