Ramification theory of valued fields
A branch of commutative algebra and number theory in which certain distinguished intermediate fields of algebraic extensions of fields equipped with a valuation are considered. Let 
be a (not necessarily finite) algebraic extension of fields, and let 
be a valuation of 
with valuation ring 
and extending a valuation 
of 
. Assume that the extension 
is normal (cf. Extension of a field) and that 
is its Galois group. The subgroup
 |
of 
is called the decomposition group of 
, and its fixed field 
the decomposition field. The subgroup
 |
of 
is called the inertia group, and its fixed field 
the inertia field. The subgroup
 |
 |
of 
is called the ramification group, and its fixed field 
the ramification field. If 
denotes the (unique) maximal ideal of 
, then the condition 
is equivalent to 
, and 
is equivalent to
 |
In number theory, also the higher ramification groups (cf. Ramified prime ideal) play a role; see [a2]. If the value group 
is a subgroup of the real numbers and 
is a real number, then the 
th ramification group is defined to be
 |
Basic properties.
Let 
denote the characteristic of the residue field 
if it is a positive prime number; otherwise, set 
. For simplicity, denote the restriction of 
to the intermediate fields again by 
. Then 
is a pro-
-group; in particular, 
if the characteristic of 
is 
. The quotient group 
of the respective value groups is a 
-group, and the extension 
of the respective residue fields is purely inseparable (cf. Separable extension). 
and 
are normal subgroups of 
, and 
is a normal subgroup of 
.
The Galois group 
of the normal separable extension 
is isomorphic to the character group 
, which is (non-canonically) isomorphic to 
if this group is finite. One has 
, and the group 
is 
-prime, i.e., no element has an order divisible by 
. Every finite quotient of the profinite group 
is 
-prime.
The Galois group 
of the normal separable extension 
is isomorphic to the Galois group of the normal extensions 
(which is 
). Furthermore, 
is separable, and 
. The extension of 
from 
to 
is unique. The extension 
is purely inseparable, and 
is a 
-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 
. Then one obtains the following additional properties: 
; 
; 
is the minimal subextension which admits a unique extension of 
to 
; 
is the maximal separable subextension of 
; and 
is the maximal of all subgroups 
of 
for which 
is 
-prime.
Absolute ramification theory.
Let 
be any field with a valuation 
, and let 
be some extension of 
to the separable-algebraic closure 
of 
. Then the intermediate fields 
are called the absolute decomposition field, the absolute inertia field and the absolute ramification field, respectively. Since all extensions of 
to 
are conjugate, that is, of the form 
for 
, it follows that these fields are independent of the choice of the extension 
, up to isomorphism over 
. The absolute ramification field is the Henselization of 
inside 
(see Henselization of a valued field); it coincides with 
if and only if the extension of 
from 
to every algebraic extension field is unique.
Tame extensions and defectless fields.
An extension 
of 
is called tamely ramified if 
is 
-prime and 
is separable. Let 
be Henselian. Then an extension of 
is called a tame extension if it is algebraic, tamely ramified and the defect of every finite subextension is trivial, that is, equal to 
. The absolute ramification field is the unique maximal tame extension of 
. If it is algebraically closed, or equivalently, if all algebraic extensions of 
are tame extensions, then 
is called a tame field; see also 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 
. Since every finite subextension of this extension has as degree a power of 
, the defect must be a power of 
. This is the content of the Ostrowski lemma. In particular, the defect is always trivial if 
, that is, if the characteristic of 
is 
.
References
| [a1] | O. Endler, "Valuation theory" , Springer (1972) |
| [a2] | J.P. Serre, "Corps locaux" , Hermann (1962) |
Ramification theory of valued fields. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ramification_theory_of_valued_fields&oldid=39688