# 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 $L \mid K$ 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) and that $G = G ( L \mid K )$ is its 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 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) play a role; see [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; 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, and the extension $Lw \mid Vw$ of the respective residue fields is purely inseparable . $G _ {V}$ and $G _ {T}$ are normal subgroups of $G _ {Z}$, and $G _ {V}$ is a 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 $G _ {T} /G _ {V}$ 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); 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 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. 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

 [a1] O. Endler, "Valuation theory" , Springer (1972) [a2] J.P. Serre, "Corps locaux" , Hermann (1962)
How to Cite This Entry:
Ramification theory of valued fields. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ramification_theory_of_valued_fields&oldid=48418
This article was adapted from an original article by F.-V. Kuhlmann (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article