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In
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
, more specifically in
local class field theory In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite res ...
, the ramification groups are a filtration of the
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
of a
local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact ...
extension, which gives detailed information on the ramification phenomena of the extension.


Ramification theory of valuations

In mathematics, the ramification theory of valuations studies the set of
extensions 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 ...
of a valuation ''v'' of a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
''K'' to an extension ''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 Value or values may refer to: Ethics and social * Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them ** Values (Western philosophy) expands the notion of value beyo ...
and let ''L'' be a
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ' ...
of ''K''. Let ''Sv'' be the set of equivalence classes of extensions of ''v'' to ''L'' and let ''G'' be the
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
of ''L'' over ''K''. Then ''G'' acts on ''Sv'' by σ 'w''nbsp;=  'w'' ∘ σ(i.e. ''w'' is a
representative Representative may refer to: Politics * Representative democracy, type of democracy in which elected officials represent a group of people * House of Representatives, legislative body in various countries or sub-national entities * Legislator, som ...
of the equivalence class 'w''nbsp;∈ ''Sv'' and 'w''is sent to the equivalence class of the
composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
of ''w'' with the automorphism ; 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 ''Gw'' of 'w'' i.e. it is the
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of ''G'' consisting of all elements that fix the equivalence class 'w''nbsp;∈ ''Sv''. Let ''mw'' denote the
maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
of ''w'' inside the
valuation ring In abstract algebra, a valuation ring is an integral domain ''D'' such that for every element ''x'' of its field of fractions ''F'', at least one of ''x'' or ''x''−1 belongs to ''D''. Given a field ''F'', if ''D'' is a subring of ''F'' suc ...
''Rw'' of ''w''. The inertia group of ''w'' is the subgroup ''Iw'' of ''Gw'' consisting of elements ''σ'' such that σ''x'' ≡ ''x'' (mod ''mw'') for all ''x'' in ''Rw''. In other words, ''Iw'' consists of the elements of the decomposition group that act trivially on the
residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a ...
of ''w''. It is a normal subgroup of ''Gw''. 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 G of a finite L/K
Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ' ...
of
local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact ...
s. We shall write w, \mathcal O_L, \mathfrak p for the valuation, the ring of integers and its maximal ideal for L. As a consequence of
Hensel's lemma In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number , then this root can be ''lifted'' to ...
, one can write \mathcal O_L = \mathcal O_K alpha/math> for some \alpha \in L where \mathcal O_K is the ring of integers of K.Neukirch (1999) p.178 (This is stronger than the
primitive element theorem In field theory, the primitive element theorem is a result characterizing the finite degree field extensions that can be generated by a single element. Such a generating element is called a primitive element of the field extension, and the extens ...
.) Then, for each integer i \ge -1, we define G_i to be the set of all s \in G that satisfies the following equivalent conditions. *(i) s operates trivially on \mathcal O_L / \mathfrak p^. *(ii) w(s(x) - x) \ge i+1 for all x \in \mathcal O_L *(iii) w(s(\alpha) - \alpha) \ge i+1. The group G_i is called ''i-th ramification group''. They form a decreasing filtration, :G_ = G \supset G_0 \supset G_1 \supset \dots \. In fact, the G_i are normal by (i) and
trivial Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense. Latin Etymology The ancient Romans used the word ''triviae'' to describe where one road split or forked ...
for sufficiently large i by (iii). For the lowest indices, it is customary to call G_0 the inertia subgroup of G because of its relation to splitting of prime ideals, while G_1 the wild inertia subgroup of G. The quotient G_0 / G_1 is called the tame quotient. The Galois group G and its subgroups G_i are studied by employing the above filtration or, more specifically, the corresponding quotients. In particular, *G/G_0 = \operatorname(l/k), where l, k are the (finite) residue fields of L, K. *G_0 = 1 \Leftrightarrow L/K is unramified. *G_1 = 1 \Leftrightarrow L/K is tamely ramified (i.e., the ramification index is prime to the residue characteristic.) The study of ramification groups reduces to the totally ramified case since one has G_i = (G_0)_i for i \ge 0. One also defines the function i_G(s) = w(s(\alpha) - \alpha), s \in G. (ii) in the above shows i_G is independent of choice of \alpha and, moreover, the study of the filtration G_i is essentially equivalent to that of i_G.Serre (1979) p.62 i_G satisfies the following: for s, t \in G, *i_G(s) \ge i + 1 \Leftrightarrow s \in G_i. *i_G(t s t^) = i_G(s). *i_G(st) \ge \min\. Fix a uniformizer \pi of L. Then s \mapsto s(\pi)/\pi induces the injection G_i/G_ \to U_/U_, i \ge 0 where U_ = \mathcal_L^\times, U_ = 1 + \mathfrak^i. (The map actually does not depend on the choice of the uniformizer.) It follows from this *G_0/G_1 is cyclic of order prime to p *G_i/G_ is a product of cyclic groups of order p. In particular, G_1 is a ''p''-group and G_0 is solvable. The ramification groups can be used to compute the different \mathfrak_ of the extension L/K and that of subextensions:Serre (1979) 4.1 Prop.4, p.64 :w(\mathfrak_) = \sum_ i_G(s) = \sum_^\infty (, G_i, - 1). If H is a normal subgroup of G, then, for \sigma \in G, i_(\sigma) = \sum_ i_G(s).Serre (1979) 4.1. Prop.3, p.63 Combining this with the above one obtains: for a subextension F/K corresponding to H, :v_F(\mathfrak_) = \sum_ i_G(s). If s \in G_i, t \in G_j, i, j \ge 1, then sts^t^ \in G_. In the terminology of
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, this can be understood to mean the Lie algebra \operatorname(G_1) = \sum_ G_i/G_ is abelian.


Example: the cyclotomic extension

The ramification groups for a cyclotomic extension K_n := \mathbf Q_p(\zeta)/\mathbf Q_p, where \zeta is a p^n-th primitive
root of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important ...
, can be described explicitly:Serre, ''Corps locaux''. Ch. IV, §4, Proposition 18 :G_s = Gal(K_n / K_e), where ''e'' is chosen such that p^ \le s < p^e.


Example: a quartic extension

Let K be the extension of generated by x_1=\sqrt. The conjugates of x1 are x2=x_2 = \sqrt, ''x''3 = −''x''1, ''x''4 = −''x''2. A little computation shows that the quotient of any two of these is a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
. Hence they all generate the same ideal; call it . \sqrt generates 2; (2)=4. Now ''x''1 − ''x''3 = 2''x''1, which is in 5. and x_1 - x_2 = \sqrt, which is in 3. Various methods show that the Galois group of ''K'' is C_4, cyclic of order 4. Also: : G_0 = G_1 = G_2 = C_4. and G_3 = G_4=(13)(24). w(\mathfrak_) = 3+3+3+1+1 = 11, so that the different \mathfrak_ = \pi^ ''x''1 satisfies ''x''4 − 4''x''2 + 2, which has discriminant 2048 = 211.


Ramification groups in upper numbering

If u is a real number \ge -1, let G_u denote G_i where ''i'' the least integer \ge u. In other words, s \in G_u \Leftrightarrow i_G(s) \ge u+1. Define \phi bySerre (1967) p.156 :\phi(u) = \int_0^u where, by convention, (G_0 : G_t) is equal to (G_ : G_0)^ if t = -1 and is equal to 1 for -1 < t \le 0.Neukirch (1999) p.179 Then \phi(u) = u for -1 \le u \le 0. It is immediate that \phi is continuous and strictly increasing, and thus has the continuous inverse function \psi defined on [-1, \infty). Define G^v = G_. G^v is then called the ''v''-th ramification group in upper numbering. In other words, G^ = G_u. Note G^ = G, G^0 = G_0. The upper numbering is defined so as to be compatible with passage to quotients:Serre (1967) p.155 if H is normal in G, then :(G/H)^v = G^v H / H for all v (whereas lower numbering is compatible with passage to subgroups.)


Herbrand's theorem

Herbrand's theorem states that the ramification groups in the lower numbering satisfy G_u H/H = (G/H)_v (for v = \phi_(u) where L/F is the subextension corresponding to H), and that the ramification groups in the upper numbering satisfy G^u H/H = (G/H)^u.Neukirch (1999) p.180Serre (1979) p.75 This allows one to define ramification groups in the upper numbering for infinite Galois extensions (such as the absolute Galois group of a local field) from the inverse system of ramification groups for finite subextensions. The upper numbering for an abelian extension is important because of the Hasse–Arf theorem. It states that if G is abelian, then the jumps in the filtration G^v are integers; i.e., G_i = G_ whenever \phi(i) is not an integer.Neukirch (1999) p.355 The upper numbering is compatible with the filtration of the norm residue group by the unit groups under the Artin isomorphism. The image of G^n(L/K) under the isomorphism : G(L/K)^ \leftrightarrow K^*/N_(L^*) is justSnaith (1994) pp.30-31 : U^n_K / (U^n_K \cap N_(L^*)) \ .


See also

*
Finite extensions of local fields In algebraic number theory, through completion, the study of ramification of a prime ideal can often be reduced to the case of local fields where a more detailed analysis can be carried out with the aid of tools such as ramification groups. In t ...


Notes


References

*B. Conrad
Math 248A. Higher ramification groups
* * * * * {{cite book , last=Snaith , first=Victor P. , title=Galois module structure , series=Fields Institute monographs , location=Providence, RI , publisher=
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
, year=1994 , isbn=0-8218-0264-X , zbl=0830.11042 Algebraic number theory