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A branch of [[Commutative algebra|commutative algebra]] and [[Number theory|number theory]] in which certain distinguished intermediate fields of algebraic extensions of fields equipped with a [[Valuation|valuation]] are considered. Let  $  L \mid  K $
be a (not necessarily finite) algebraic extension of fields, and let  $  w $
be a valuation of  $  L $
with valuation ring  $  {\mathcal O} _ {w} $
and extending a valuation  $  v $
of  $  K $. 
Assume that the extension  $  L \mid  K $
is normal (cf. [[Extension of a field|Extension of a field]]) and that  $  G = G ( L \mid  K ) $
is its [[Galois group|Galois group]]. The subgroup

$$ 
G _ {Z} = \left \{ {\sigma \in G } : {w ( \sigma a ) = w ( a )  \textrm{ for  all  }  a \in L } \right \}
$$

of  $  G $
is called the decomposition group of  $  w \mid  v $, 
and its fixed field  $  Z $
the decomposition field. The subgroup

$$ 
G _ {T} = \left \{ {\sigma \in G } : {w ( \sigma a - a ) > 0  \textrm{ for  all  }  a \in {\mathcal O} _ {w} } \right \}
$$

of  $  G _ {Z} $
is called the inertia group, and its fixed field  $  T $
the inertia field. The subgroup

$$ 
G _ {V} =
$$

$$ 
= 
\left \{ {\sigma \in G } : {w ( \sigma a - a ) > w ( a )  \textrm{ for  all  }  a \in L,  a \neq 0 } \right \}
$$

of  $  G _ {T} $
is called the ramification group, and its fixed field  $  V $
the ramification field. If  $  {\mathcal M} _ {w} $
denotes the (unique) maximal [[Ideal|ideal]] of  $  {\mathcal O} _ {w} $, 
then the condition  $  w ( \sigma a - a ) > 0 $
is equivalent to  $  \sigma a - a \in {\mathcal M} _ {w} $, 
and  $  w ( \sigma a - a ) > w ( a ) $
is equivalent to

$$ 
{
\frac{\sigma a }{a}
 } - 1 \in {\mathcal M} _ {w} .
$$

In number theory, also the higher ramification groups (cf. [[Ramified prime ideal|Ramified prime ideal]]) play a role; see [[#References|[a2]]]. If the value group  $  wL $
is a subgroup of the real numbers and  $  s \geq  - 1 $
is a real number, then the  $  s $
th ramification group is defined to be

$$ 
\left \{ {\sigma \in G } : {w ( \sigma a - a ) \geq  s + 1  \textrm{ for  all  }  a \in {\mathcal O} _ {w} } \right \} .
$$

==Basic properties.==
Let  $  p $
denote the characteristic of the residue field  $  Lw $
if it is a positive prime number; otherwise, set  $  p = 1 $. 
For simplicity, denote the restriction of  $  w $
to the intermediate fields again by  $  w $. 
Then  $  G _ {V} $
is a [[Pro-p group|pro- $  p $-
group]]; in particular,  $  L = V $
if the characteristic of  $  Lw $
is  $  0 $. 
The quotient group  $  wL/wV $
of the respective value groups is a [[P-group| $  p $-
group]], and the extension  $  Lw \mid  Vw $
of the respective residue fields is [[Purely inseparable extension|purely inseparable]] .  $  G _ {V} $
and  $  G _ {T} $
are normal subgroups of  $  G _ {Z} $, 
and  $  G _ {V} $
is a [[Normal subgroup|normal subgroup]] of  $  G _ {T} $.

The [[Galois group]]  $  G _ {T} /G _ {V} $
of the normal separable extension  $  V \mid  T $
is isomorphic to the character group  $  { \mathop{\rm Hom} } ( wL/vK,Lw  ^  \times  ) $, 
which is (non-canonically) isomorphic to  $  wV/wT $
if this group is finite. One has  $  Vw = Tw $, 
and the group  $  wV/wT $
is  $  p $-
prime, i.e., no element has an order divisible by  $  p $. 
Every finite quotient of the [[Profinite group|profinite group]]  $  G _ {T} /G _ {V} $
is  $  p $-
prime.

The Galois group  $  G _ {Z} /G _ {T} $
of the normal separable extension  $  T \mid  Z $
is isomorphic to the Galois group of the normal extensions  $  Lw \mid  Kv $(
which is  $  Tw \mid  Zw $). 
Furthermore,  $  Tw \mid  Zw $
is separable, and  $  wT = wZ $. 
The extension of  $  w $
from  $  Z $
to  $  L $
is unique. The extension  $  Zw \mid  Kv $
is purely inseparable, and  $  wZ/vK $
is a  $  p $-
group.

For many applications, it is more convenient to define the decomposition, inertia and ramification field to be the fixed field of the corresponding group in the maximal separable subextension of  $  L \mid  K $. 
Then one obtains the following additional properties:  $  wZ = vK $; 
$  Zw = Kv $; 
$  Z $
is the minimal subextension which admits a unique extension of  $  w $
to  $  L $; 
$  Tw \mid  Kv $
is the maximal separable subextension of  $  Lw \mid  Kv  $; 
and  $  wV $
is the maximal of all subgroups  $  \Gamma $
of  $  wL $
for which  $  \Gamma/vK $
is  $  p $-
prime.

==Absolute ramification theory.==
Let  $  K $
be any field with a valuation  $  v $, 
and let  $  v  ^ {s} $
be some extension of  $  v $
to the separable-algebraic closure  $  K  ^ {s} $
of  $  K $. 
Then the intermediate fields  $  Z,T,V $
are called the absolute decomposition field, the absolute inertia field and the absolute ramification field, respectively. Since all extensions of  $  v $
to  $  K  ^ {s} $
are conjugate, that is, of the form  $  v \circ \sigma $
for  $  \sigma \in G ( K  ^ {s} \mid  K ) $, 
it follows that these fields are independent of the choice of the extension  $  v  ^ {s} $, 
up to isomorphism over  $  K $. 
The absolute ramification field is the Henselization of  $  ( K,v ) $
inside  $  ( K  ^ {s} ,v  ^ {s} ) $(
see [[Henselization of a valued field|Henselization of a valued field]]); it coincides with  $  K $
if and only if the extension of  $  v $
from  $  K $
to every algebraic extension field is unique.

==Tame extensions and defectless fields.==
An extension  $  ( L,w ) $
of  $  ( K,v ) $
is called tamely ramified if  $  wL/vK $
is  $  p $-
prime and  $  Lw \mid  Kv $
is separable. Let  $  ( K,v ) $
be Henselian. Then an extension of  $  ( K,v ) $
is called a tame extension if it is algebraic, tamely ramified and the [[Defect|defect]] of every finite subextension is trivial, that is, equal to  $  1 $. 
The absolute ramification field is the unique maximal tame extension of  $  ( K,v ) $. 
If it is algebraically closed, or equivalently, if all algebraic extensions of  $  ( K,v ) $
are tame extensions, then  $  ( K,v ) $
is called a tame field; see also [[Model theory of valued fields|Model theory of valued fields]]. From the fact that every finite subextension in the absolute ramification field is defectless it follows that a non-trivial defect can only appear between the absolute ramification field and the algebraic closure of  $  K $. 
Since every finite subextension of this extension has as degree a power of  $  p $, 
the defect must be a power of  $  p $. 
This is the content of the Ostrowski lemma. In particular, the defect is always trivial if  $  p = 1 $, 
that is, if the characteristic of  $  Kv $
is  $  0 $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  O. Endler,   "Valuation theory" , Springer  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.P. Serre,   "Corps locaux" , Hermann  (1962)</TD></TR></table>