Approximation of elements in henselizations

TL;DR

This paper generalizes the approximation of elements in henselizations for valued fields of higher rank, extending known results from rank 1 and exploring implications for algebraic extensions.

Contribution

It provides a new method to approximate elements in henselizations of higher rank valued fields from within the base field.

Findings

Elements in henselizations of higher rank valued fields can be approximated from the base field.

Approximation properties influence the linear disjointness of algebraic extensions.

The results unify and extend classical approximation results for rank 1 fields.

Abstract

For valued fields $K$ of rank higher than 1, we describe how elements in the henselization $K^h$ of $K$ can be approximated from within $K$; our result is a handy generalization of the well-known fact that in rank 1, all of these elements lie in the completion of $K$. We apply the result to show that if an element $z$ algebraic over $K$ can be approximated from within $K$ in the same way as an element in $K^h$, then $K(z)$ is not linearly disjoint from $K^h$ over $K$.