# Immediately algebraically closed fields

## TL;DR

This paper investigates the relationship between two classes of fields, IAC and VAC, defined via valuation theory, providing a counterexample to Hong's question and identifying conditions under which they coincide.

## Contribution

The paper constructs a counterexample showing IAC and VAC are not always equal and explores conditions where they do coincide, advancing understanding of valuation-theoretic field classes.

## Key findings

- Counterexample disproves Hong's question
- IAC and VAC are not always equal
- Certain conditions ensure IAC and VAC coincide

## Abstract

We consider two overlapping classes of fields, IAC and VAC, which are defined using valuation theory but which do not involve a distinguished valuation. Rather, each class is defined by a condition that quantifies over all possible valuations on the field. In his thesis, Hong asked whether these two classes are equal (Hong, 2013, Question 5.6.8). In this paper, we give an example that negatively answers Hong's question. We also explore several situations in which the equivalence does hold with an additional assumption, including the case where every $K'\equiv K$ is IAC.