# 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.

## Galois case

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*by σ[

_{v}*w*] = [

*w*∘ σ] (i.e.

*w*is a representative of the equivalence class [

*w*] ∈

*S*and [

_{v}*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*of [

_{w}*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*of

_{w}*w*. The

**inertia group of**is the subgroup

*w**I*of

_{w}*G*consisting of elements

_{w}*σ*such that σ

*x*≡

*x*(mod

*m*) for all

_{w}*x*in

*R*. In other words,

_{w}*I*consists of the elements of the decomposition group that act trivially on the residue field of

_{w}*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*).

## See also

## References

- Fröhlich, A.; Taylor, M.J. (1991).
*Algebraic number theory*. Cambridge studies in advanced mathematics.**27**. Cambridge University Press. ISBN 0-521-36664-X. Zbl 0744.11001. - Zariski, Oscar; Samuel, Pierre (1976) [1960].
*Commutative algebra, Volume II*. Graduate Texts in Mathematics.**29**. New York, Heidelberg: Springer-Verlag. Chapter VI. ISBN 978-0-387-90171-8. Zbl 0322.13001.