Dense subfields of henselian fields, and integer parts

TL;DR

This paper investigates dense subfields within henselian valued fields of residue characteristic zero, exploring conditions for transcendental extensions, and examines properties of integer parts in ordered fields and related structures.

Contribution

It introduces new conditions for the existence of dense henselian subfields and analyzes integer parts in ordered fields, providing novel examples and counterexamples.

Findings

Every henselian valued field of residue characteristic 0 has a proper dense subfield.

Conditions are identified for such subfields to be transcendental and henselian.

Examples of real closed fields larger than their integer parts are provided.

Abstract

We show that every henselian valued field $L$ of residue characteristic 0 admits a proper subfield $K$ which is dense in $L$. We present conditions under which this can be taken such that $L|K$ is transcendental and $K$ is henselian. These results are of interest for the investigation of integer parts of ordered fields. We present examples of real closed fields which are larger than the quotient fields of all their integer parts. Finally, we give rather simple examples of ordered fields that do not admit any integer part and of valued fields that do not admit any subring which is an additive complement of the valuation ring.