# Ramification group

In number theory, more specifically in local class field theory, the ramification groups are a filtration 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, the ramification theory of valuations studies the set of extensions of a valuation v of a field 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 and let L be a finite Galois extension of K. Let S$_v$ be the set of equivalence 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] = [w ∘ σ] (i.e. w is a representative of the equivalence class [w] ∈ S$_v$ and [w] is sent to the equivalence class of the composition of w with the automorphism σ : L → L; 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 G$_w$ of [w], i.e. it is the subgroup of G consisting of all elements that fix the equivalence class [w] ∈ 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 σx ≡ 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 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 $G$ of a finite $L/K$ Galois extension of local fields. 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, one can write ${\mathcal {O}}_{L}={\mathcal {O}}_{K}[\alpha ]$ for some $\alpha \in L$ where ${\mathcal {O}}_{K}$ is the ring of integers of $K$. (This is stronger than the primitive element theorem.) Then, for each integer $i\geq -1$, we define $G_{i}$ to be the set of all $s\in G$ that satisfies the following equivalent conditions.
* (i) $s$ operates trivially on ${\mathcal {O}}_{L}/{\mathfrak {p}}^{i+1}.$
* (ii) $w(s(x)-x)\geq i+1$ for all $x\in {\mathcal {O}}_{L}$
* (iii) $w(s(\alpha )-\alpha )\geq i+1.$
The group $G_{i}$ is called $i$-th ramification group. They form a decreasing filtration,
    $G_{-1}=G\supset G_{0}\supset G_{1}\supset \dots \{*\}.$
In fact, the $G_{i}$ are normal by (i) and trivial 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,
* $G/G_{0}=\operatorname {Gal} (l/k),$ 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 $i\geq 0$.
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 $i_{G}$. $i_{G}$ satisfies the following: for $s,t\in G$,
* $i_{G}(s)\geq i+1\Leftrightarrow s\in G_{i}.$
* $i_{G}(tst^{-1})=i_{G}(s).$
* $i_{G}(st)\geq \min\{i_{G}(s),i_{G}(t)\}.$
Fix a uniformizer $\pi$ of $L$. Then $s\mapsto s(\pi )/\pi$ induces the injection $G_{i}/G_{i+1}\to U_{L,i}/U_{L,i+1},i\geq 0$ where $U_{L,0}={\mathcal {O}}_{L}^{\times },U_{L,i}=1+{\mathfrak {p}}^{i}$. (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 $p$
* $G_{i}/G_{i+1}$ 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 ${\mathfrak {D}}_{L/K}$ of the extension $L/K$ and that of subextensions:
    $w({\mathfrak {D}}_{L/K})=\sum _{s\neq 1}i_{G}(s)=\sum _{i=0}^{\infty }(|G_{i}|-1).$
If $H$ is a normal subgroup of $G$, then, for $\sigma \in G$, $i_{G/H}(\sigma )={1 \over e_{L/K}}\sum _{s\mapsto \sigma }i_{G}(s)$.
Combining this with the above one obtains: for a subextension $F/K$ corresponding to $H$,
    $v_{F}({\mathfrak {D}}_{F/K})={1 \over e_{L/F}}\sum _{s\not \in H}i_{G}(s).$
If $s\in G_{i},t\in G_{j},i,j\geq 1$, then $sts^{-1}t^{-1}\in G_{i+j+1}$. In the terminology of Lazard, this can be understood to mean the Lie algebra $\operatorname {gr} (G_{1})=\sum _{i\geq 1}G_{i}/G_{i+1}$ 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, can be described explicitly:
    $G_{s}=\operatorname {Gal} (K_{n}/K_{e}),$
where e is chosen such that $p^{e-1}\leq s<p^{e}$.

### Example: a quartic extension

Let K be the extension of Q$_2$ generated by $x_{1}={\sqrt {2+{\sqrt {2}}}}$. The conjugates of $x_{1}$ are $x_{2}={\sqrt {2-{\sqrt {2}}}}$, $x_{3}=-x_{1}$, $x_{4}=-x_{2}$.
A little computation shows that the quotient of any two of these is a unit. Hence they all generate the same ideal; call it π. ${\sqrt {2}}$ generates π$^2$; (2)=π$^4$.
Now $x_{1}-x_{3}=2x_{1}$, which is in π$^5$.
and $x_{1}-x_{2}={\sqrt {4-2{\sqrt {2}}}},$ which is in π$^3$.
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 $G_{3}=G_{4}=(13)(24).$
$w({\mathfrak {D}}_{K/Q_{2}})=3+3+3+1+1=11,$ so that the different ${\mathfrak {D}}_{K/Q_{2}}=\pi ^{11}$
$x_{1}$ satisfies X$^4$ − 4X$^2$ + 2, which has discriminant 2048 = 2$^{11}$.

## Ramification groups in upper numbering

If $u$ is a real number $\geq -1$, let $G_{u}$ denote $G_{i}$ where i the least integer $\geq u$. In other words, $s\in G_{u}\Leftrightarrow i_{G}(s)\geq u+1.$ Define $\phi$ by
    $\phi (u)=\int _{0}^{u}{dt \over (G_{0}:G_{t})}$
where, by convention, $(G_{0}:G_{t})$ is equal to $(G_{-1}:G_{0})^{-1}$ if $t=-1$ and is equal to $1$ for $-1<t\leq 0$. Then $\phi (u)=u$ for $-1\leq u\leq 0$. It is immediate that $\phi$ is continuous and strictly increasing, and thus has the continuous inverse function $\psi$ defined on $[-1,\infty )$. Define $G^{v}=G_{\psi (v)}$. $G^{v}$ is then called the v-th ramification group in upper numbering. In other words, $G^{\phi (u)}=G_{u}$. Note $G^{-1}=G,G^{0}=G_{0}$. The upper numbering is defined so as to be compatible with passage to quotients: 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 $v=\phi _{L/F}(u)$ where $L/F$ is the subextension corresponding to $H$), and that the ramification groups in the upper numbering satisfy $G^{u}H/H=(G/H)^{u}$. 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., $G_{i}=G_{i+1}$ whenever $\phi (i)$ is not an integer.
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
    $G(L/K)^{\mathrm {ab} }\leftrightarrow K^{*}/N_{L/K}(L^{*})$
is just
    $U_{K}^{n}/(U_{K}^{n}\cap N_{L/K}(L^{*}))\ .$