# Séminaires

**On the structure of immediate valued function fields**

**Le : **05/07/2013 14h00

**Par : **Franz-Viktor Kuhlmann (université de Sakkatoon, Canada)

**Lieu : **I 001

**Lien web : **

**Résumé : **Elimination of wild ramification is an important part of local
uniformization. In the particular case of an immediate valued function
field F|K of transcendence degree 1 it means to find an element x in F
such that F lies in the henselization of K(x). (An extension of valued
fields is called immediate if value group and residue field do not
change.)
This task is trivial if the characteristic of the residue field is 0, and
highly nontrivial in the positive characteristic case. In order to prove a
theorem of "Henselian Rationality" over suitable valued ground fields K,
we need quite a bit of technical preparation, which can be seen as a
continuation of Kaplansky's important paper "Maximal fields with
valuations I", Duke Math. Journal 9 (1942), 303-321.
In the first step of the proof, we find the element x in the henselization
F^h of F. In this step it is essential to know the degree of the extension
K(x)^h|K(f(x))^h for polynomials f.
In the second step of the proof, once x is found in F^h, we try to replace
it by an element y in F. Here, we need to know the degree of the extension
K(x)^h|K(y)^h for arbitrary elements in K(x)^h.
This step can be seen as a special case of "dehenselization",
corresponding to the notion of "decompletion" that appears in Temkin's
work. Dehenselization in general states that if there is a finite
extension F' of F within its henselization F^h such that F'|K admits local
uniformization, then so does F|K. Whether this is true is an important
open problem. The above special case and Temkin's "decompletion" seem to
indicate that the problem could be solved to the affirmative.
One tool in the proofs is to use ramification theory to reduce the problem
to considering Galois extensions of degree p, by extending the ground
field. Then in the end, henselian rationality has to be pulled down
through such ("tame") extensions. This too can be solved within the
described framework where criteria are given for the degree of the
extension K(x)^h|K(f(x))^h to be 1. Concretely, it can be shown that the
trace of a well chosen henselian generator "upstairs" is a henselian
generator "downstairs".