# Elimination of Ramification II: Henselian Rationality

## TL;DR

This paper proves that certain valued function fields over tame fields can be embedded into henselizations of rational function fields, eliminating ramification and impacting local uniformization and model theory in positive and mixed characteristic.

## Contribution

It establishes that immediate valued algebraic function fields of transcendence degree one over tame fields are contained in henselizations of rational function fields, generalizing previous results.

## Key findings

- Elimination of ramification in valued function fields over tame fields
- Embedding of such fields into henselizations of rational functions
- Applications to local uniformization and model theory in positive/mixed characteristic

## Abstract

We prove in arbitrary characteristic that an immediate valued algebraic function field $F$ of transcendence degree 1 over a tame field $K$ is contained in the henselization of $K(x)$ for a suitably chosen $x\in F$. This eliminates ramification in such valued function fields. We give generalizations of this result, relaxing the assumption on $K$. Our theorems have important applications to local uniformization and to the model theory of valued fields in positive and mixed characteristic.