In
number theory, more specifically in
local class field theory, the ramification groups are a
filtration
Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...
of the
Galois group of a
local field extension, which gives detailed information on the
ramification phenomena of the extension.
Ramification theory of valuations
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the ramification theory of valuations studies the set of
extensions of a
valuation ''v'' of a
field ''K'' to an
extension
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* E ...
''L'' of ''K''. It is a generalization of the ramification theory of Dedekind domains.
The structure of the set of extensions is known better when ''L''/''K'' is
Galois.
Decomposition group and inertia group
Let (''K'', ''v'') be a
valued field and let ''L'' be a
finite Galois extension of ''K''. Let ''S
v'' be the set of
equivalence
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
*Equivalence class (music)
*''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*'' Equival ...
classes of extensions of ''v'' to ''L'' and let ''G'' be the
Galois group of ''L'' over ''K''. Then ''G'' acts on ''S
v'' by σ
'w''nbsp;=
'w'' ∘ σ(i.e. ''w'' is a
representative of the equivalence class
'w''nbsp;∈ ''S
v'' and
'w''is sent to the equivalence class of the
composition of ''w'' with the
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
; this is independent of the choice of ''w'' in
'w''. In fact, this action is
transitive.
Given a fixed extension ''w'' of ''v'' to ''L'', the decomposition group of ''w'' is the
stabilizer subgroup
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
''G
w'' of
'w'' i.e. it is the
subgroup of ''G'' consisting of all elements that fix the equivalence class
'w''nbsp;∈ ''S
v''.
Let ''m
w'' denote the
maximal ideal of ''w'' inside the
valuation ring ''R
w'' of ''w''. The inertia group of ''w'' is the subgroup ''I
w'' of ''G
w'' consisting of elements ''σ'' such that σ''x'' ≡ ''x'' (mod ''m
w'') for all ''x'' in ''R
w''. In other words, ''I
w'' consists of the elements of the decomposition group that
act trivially on the
residue field of ''w''. It is a
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
of ''G
w''.
The
reduced ramification index ''e''(''w''/''v'') is independent of ''w'' and is denoted ''e''(''v''). Similarly, the
relative degree ''f''(''w''/''v'') is also independent of ''w'' and is denoted ''f''(''v'').
Ramification groups in lower numbering
Ramification groups are a refinement of the Galois group
of a finite
Galois extension of
local fields. We shall write
for the valuation, the ring of integers and its maximal ideal for
. As a consequence of
Hensel's lemma, one can write