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''ramification deficiency''
 
''ramification deficiency''
  
An invariant of finite extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d1100801.png" /> of fields equipped with a [[Valuation|valuation]] (cf. also [[Extension of a field|Extension of a field]]). If a valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d1100802.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d1100803.png" /> is the unique extension of a valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d1100804.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d1100805.png" />, then the defect (or ramification deficiency) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d1100806.png" /> is defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d1100807.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d1100808.png" /> is the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d1100809.png" /> (i.e., the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008010.png" /> as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008011.png" />-vector space), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008012.png" /> is the ramification index and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008013.png" /> is the inertia degree. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008014.png" /> denote the respective value groups and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008015.png" /> the respective residue fields. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008016.png" /> admits several extensions to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008017.png" />, the defect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008018.png" /> can be defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008020.png" /> is the number of distinct extensions, provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008021.png" /> is normal (since in that case the ramification index and the inertia degree are the same for all extensions; cf. also [[Normal extension|Normal extension]]).
+
An invariant of finite extensions $  L \mid  K $
 +
of fields equipped with a [[Valuation|valuation]] (cf. also [[Extension of a field|Extension of a field]]). If a valuation $  w $
 +
on $  L $
 +
is the unique extension of a valuation $  v $
 +
on $  K $,  
 +
then the defect (or ramification deficiency) $  d = d ( w \mid  v ) $
 +
is defined by the formula $  [ L:K ] = d \cdot e \cdot f $,  
 +
where $  [ L:K ] $
 +
is the degree of $  L \mid  K $(
 +
i.e., the dimension of $  L $
 +
as a $  K $-
 +
vector space), $  e = e ( w \mid  v ) = ( wL:vK ) $
 +
is the ramification index and $  f = f ( w \mid  v ) = [ Lw:Kv ] $
 +
is the inertia degree. Here, $  wL,vK $
 +
denote the respective value groups and $  Lw,Kv $
 +
the respective residue fields. If $  v $
 +
admits several extensions to $  L $,  
 +
the defect $  d = d ( w \mid  v ) $
 +
can be defined by $  [ L:K ] = d \cdot e \cdot f \cdot g $,  
 +
where $  g $
 +
is the number of distinct extensions, provided that $  L \mid  K $
 +
is normal (since in that case the ramification index and the inertia degree are the same for all extensions; cf. also [[Normal extension|Normal extension]]).
  
In the above cases, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008024.png" /> are divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008025.png" />. The defect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008026.png" /> is either equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008027.png" /> or is a power of the characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008029.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008030.png" />; otherwise, it is always equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008031.png" /> (this is the Ostrowski lemma, cf. [[Ramification theory of valued fields|Ramification theory of valued fields]]).
+
In the above cases, $  e $,  
 +
$  f $,  
 +
$  g $
 +
are divisors of $  [ L:K ] $.  
 +
The defect d $
 +
is either equal to $  1 $
 +
or is a power of the characteristic $  p $
 +
of $  Kv $
 +
if  $  p > 0 $;  
 +
otherwise, it is always equal to $  1 $(
 +
this is the Ostrowski lemma, cf. [[Ramification theory of valued fields|Ramification theory of valued fields]]).
  
 
==Henselian defect.==
 
==Henselian defect.==
To avoid considering several valuations and to have a defect available for arbitrary finite extensions, one can pass to a Henselization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008033.png" /> and a Henselization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008034.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008035.png" /> inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008036.png" /> (cf. [[Henselization of a valued field|Henselization of a valued field]]). The Henselian defect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008037.png" /> is then defined to be the defect of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008038.png" /> (by the definition of the Henselization, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008039.png" /> is the unique extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008040.png" />). In the above cases, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008041.png" />.
+
To avoid considering several valuations and to have a defect available for arbitrary finite extensions, one can pass to a Henselization $  ( L  ^ {H} ,w  ^ {H} ) $
 +
of $  ( L,w ) $
 +
and a Henselization $  ( K  ^ {H} ,v  ^ {H} ) $
 +
of $  ( K,v ) $
 +
inside $  ( L  ^ {H} ,w  ^ {H} ) $(
 +
cf. [[Henselization of a valued field|Henselization of a valued field]]). The Henselian defect $  \delta ( w \mid  v ) $
 +
is then defined to be the defect of $  w  ^ {H} \mid  v  ^ {H} $(
 +
by the definition of the Henselization, $  w  ^ {H} $
 +
is the unique extension of $  v  ^ {H} $).  
 +
In the above cases, $  \delta ( w \mid  v ) = d ( w \mid  v ) $.
  
 
==Defectless fields.==
 
==Defectless fields.==
A field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008042.png" /> with a valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008043.png" /> is called a defectless field if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008044.png" /> for every finite normal extension. This holds if and only if the Henselian defect is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008045.png" /> for every finite extension. It follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008046.png" /> is a defectless field if and only some Henselization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008047.png" /> is (or equivalently, all Henselizations are).
+
A field $  K $
 +
with a valuation $  v $
 +
is called a defectless field if $  d ( w \mid  v ) = 1 $
 +
for every finite normal extension. This holds if and only if the Henselian defect is equal to $  1 $
 +
for every finite extension. It follows that $  ( K,v ) $
 +
is a defectless field if and only some Henselization of $  ( K,v ) $
 +
is (or equivalently, all Henselizations are).
  
It follows from the Ostrowski lemma that all valued fields with residue field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008048.png" /> are defectless fields. Also, valued fields of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008049.png" /> with value group isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008050.png" /> are defectless. Combining both facts, it is shown that finitely ramified fields, and hence also fields with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008051.png" />-valuations (see [[P-adically closed field|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008052.png" />-adically closed field]]), are defectless.
+
It follows from the Ostrowski lemma that all valued fields with residue field of characteristic 0 $
 +
are defectless fields. Also, valued fields of characteristic 0 $
 +
with value group isomorphic to $  \mathbf Z $
 +
are defectless. Combining both facts, it is shown that finitely ramified fields, and hence also fields with $  p $-
 +
valuations (see [[P-adically closed field| $  p $-
 +
adically closed field]]), are defectless.
  
 
If a valued field does not admit any non-trivial immediate extension (cf. also [[Valuation|Valuation]]), then it is called a maximal valued field. Fields of formal [[Laurent series|Laurent series]] with their canonical valuations are maximal. Every maximal valued field is defectless.
 
If a valued field does not admit any non-trivial immediate extension (cf. also [[Valuation|Valuation]]), then it is called a maximal valued field. Fields of formal [[Laurent series|Laurent series]] with their canonical valuations are maximal. Every maximal valued field is defectless.
  
 
==Fundamental inequality.==
 
==Fundamental inequality.==
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008053.png" /> are all extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008054.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008055.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008056.png" />, then one has the fundamental inequality
+
If $  w _ {1} \dots w _ {m} $
 +
are all extensions of $  v $
 +
from $  K $
 +
to $  L $,  
 +
then one has the fundamental inequality
  
<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/d/d110/d110080/d11008057.png" /></td> </tr></table>
+
$$
 +
[ L:K ] \geq  \sum _ {i = 1 } ^ { m }  e ( w _ {i} \mid  v ) \cdot f ( w _ {i} \mid  w ) .
 +
$$
  
This is an equality for every finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008058.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008059.png" /> is defectless. Also, in general it can be written as an equality. For this, choose Henselizations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008060.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008062.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008063.png" /> inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008064.png" />. It is known that
+
This is an equality for every finite $  L \mid  K $
 +
if and only if $  ( K,v ) $
 +
is defectless. Also, in general it can be written as an equality. For this, choose Henselizations $  ( L ^ {H _ {i} } ,w ^ {H _ {i} } ) $
 +
of $  ( L,w _ {i} ) $
 +
and $  ( K ^ {H _ {i} } ,v ^ {H _ {i} } ) $
 +
of $  ( K,v ) $
 +
inside $  ( L ^ {H _ {i} } ,w ^ {H _ {i} } ) $.  
 +
It is known that
  
<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/d/d110/d110080/d11008065.png" /></td> </tr></table>
+
$$
 +
[ L:K ] = \sum _ {i = 1 } ^ { m }  [ L ^ {H _ {i} } :K ^ {H _ {i} } ] .
 +
$$
  
 
Further,
 
Further,
  
<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/d/d110/d110080/d11008066.png" /></td> </tr></table>
+
$$
 +
[ L ^ {H _ {i} } :K ^ {H _ {i} } ] =
 +
$$
  
<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/d/d110/d110080/d11008067.png" /></td> </tr></table>
+
$$
 +
=  
 +
d ( w ^ {H _ {i} } \mid  v ^ {H _ {i} } ) \cdot e ( w ^ {H _ {i} } \mid  v ^ {H _ {i} } ) \cdot f ( w ^ {H _ {i} } \mid  v ^ {H _ {i} } ) .
 +
$$
  
Since Henselizations are immediate extensions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008069.png" />. By definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110080/d11008070.png" />. Hence,
+
Since Henselizations are immediate extensions, $  e ( w ^ {H _ {i} } \mid  v ^ {H _ {i} } ) = e ( w \mid  v ) $
 +
and $  f ( w ^ {H _ {i} } \mid  v ^ {H _ {i} } ) = f ( w \mid  v ) $.  
 +
By definition, $  d ( w ^ {H _ {i} } \mid  v ^ {H _ {i} } ) = \delta ( w _ {i} \mid  v ) $.  
 +
Hence,
  
<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/d/d110/d110080/d11008071.png" /></td> </tr></table>
+
$$
 +
[ L:K ] = \sum _ {i = 1 } ^ { m }  \delta ( w _ {i} \mid  v ) \cdot e ( w _ {i} \mid  v ) \cdot f ( w _ {i} \mid  w ) .
 +
$$
  
 
Several other notions of defects were introduced. For a detailed theory of the defect, see [[#References|[a1]]]. See also [[Valued function field|Valued function field]].
 
Several other notions of defects were introduced. For a detailed theory of the defect, see [[#References|[a1]]]. See also [[Valued function field|Valued function field]].

Latest revision as of 17:32, 5 June 2020


ramification deficiency

An invariant of finite extensions $ L \mid K $ of fields equipped with a valuation (cf. also Extension of a field). If a valuation $ w $ on $ L $ is the unique extension of a valuation $ v $ on $ K $, then the defect (or ramification deficiency) $ d = d ( w \mid v ) $ is defined by the formula $ [ L:K ] = d \cdot e \cdot f $, where $ [ L:K ] $ is the degree of $ L \mid K $( i.e., the dimension of $ L $ as a $ K $- vector space), $ e = e ( w \mid v ) = ( wL:vK ) $ is the ramification index and $ f = f ( w \mid v ) = [ Lw:Kv ] $ is the inertia degree. Here, $ wL,vK $ denote the respective value groups and $ Lw,Kv $ the respective residue fields. If $ v $ admits several extensions to $ L $, the defect $ d = d ( w \mid v ) $ can be defined by $ [ L:K ] = d \cdot e \cdot f \cdot g $, where $ g $ is the number of distinct extensions, provided that $ L \mid K $ is normal (since in that case the ramification index and the inertia degree are the same for all extensions; cf. also Normal extension).

In the above cases, $ e $, $ f $, $ g $ are divisors of $ [ L:K ] $. The defect $ d $ is either equal to $ 1 $ or is a power of the characteristic $ p $ of $ Kv $ if $ p > 0 $; otherwise, it is always equal to $ 1 $( this is the Ostrowski lemma, cf. Ramification theory of valued fields).

Henselian defect.

To avoid considering several valuations and to have a defect available for arbitrary finite extensions, one can pass to a Henselization $ ( L ^ {H} ,w ^ {H} ) $ of $ ( L,w ) $ and a Henselization $ ( K ^ {H} ,v ^ {H} ) $ of $ ( K,v ) $ inside $ ( L ^ {H} ,w ^ {H} ) $( cf. Henselization of a valued field). The Henselian defect $ \delta ( w \mid v ) $ is then defined to be the defect of $ w ^ {H} \mid v ^ {H} $( by the definition of the Henselization, $ w ^ {H} $ is the unique extension of $ v ^ {H} $). In the above cases, $ \delta ( w \mid v ) = d ( w \mid v ) $.

Defectless fields.

A field $ K $ with a valuation $ v $ is called a defectless field if $ d ( w \mid v ) = 1 $ for every finite normal extension. This holds if and only if the Henselian defect is equal to $ 1 $ for every finite extension. It follows that $ ( K,v ) $ is a defectless field if and only some Henselization of $ ( K,v ) $ is (or equivalently, all Henselizations are).

It follows from the Ostrowski lemma that all valued fields with residue field of characteristic $ 0 $ are defectless fields. Also, valued fields of characteristic $ 0 $ with value group isomorphic to $ \mathbf Z $ are defectless. Combining both facts, it is shown that finitely ramified fields, and hence also fields with $ p $- valuations (see $ p $- adically closed field), are defectless.

If a valued field does not admit any non-trivial immediate extension (cf. also Valuation), then it is called a maximal valued field. Fields of formal Laurent series with their canonical valuations are maximal. Every maximal valued field is defectless.

Fundamental inequality.

If $ w _ {1} \dots w _ {m} $ are all extensions of $ v $ from $ K $ to $ L $, then one has the fundamental inequality

$$ [ L:K ] \geq \sum _ {i = 1 } ^ { m } e ( w _ {i} \mid v ) \cdot f ( w _ {i} \mid w ) . $$

This is an equality for every finite $ L \mid K $ if and only if $ ( K,v ) $ is defectless. Also, in general it can be written as an equality. For this, choose Henselizations $ ( L ^ {H _ {i} } ,w ^ {H _ {i} } ) $ of $ ( L,w _ {i} ) $ and $ ( K ^ {H _ {i} } ,v ^ {H _ {i} } ) $ of $ ( K,v ) $ inside $ ( L ^ {H _ {i} } ,w ^ {H _ {i} } ) $. It is known that

$$ [ L:K ] = \sum _ {i = 1 } ^ { m } [ L ^ {H _ {i} } :K ^ {H _ {i} } ] . $$

Further,

$$ [ L ^ {H _ {i} } :K ^ {H _ {i} } ] = $$

$$ = d ( w ^ {H _ {i} } \mid v ^ {H _ {i} } ) \cdot e ( w ^ {H _ {i} } \mid v ^ {H _ {i} } ) \cdot f ( w ^ {H _ {i} } \mid v ^ {H _ {i} } ) . $$

Since Henselizations are immediate extensions, $ e ( w ^ {H _ {i} } \mid v ^ {H _ {i} } ) = e ( w \mid v ) $ and $ f ( w ^ {H _ {i} } \mid v ^ {H _ {i} } ) = f ( w \mid v ) $. By definition, $ d ( w ^ {H _ {i} } \mid v ^ {H _ {i} } ) = \delta ( w _ {i} \mid v ) $. Hence,

$$ [ L:K ] = \sum _ {i = 1 } ^ { m } \delta ( w _ {i} \mid v ) \cdot e ( w _ {i} \mid v ) \cdot f ( w _ {i} \mid w ) . $$

Several other notions of defects were introduced. For a detailed theory of the defect, see [a1]. See also Valued function field.

References

[a1] F.-V. Kuhlmann, "Valuation theory of fields, abelian groups and modules" , Algebra, Logic and Applications , Gordon&Breach (to appear)
How to Cite This Entry:
Defect(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Defect(2)&oldid=17106
This article was adapted from an original article by F.-V. Kuhlmann (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article