Quasi-valuations - topology and the weak approximation theorem

TL;DR

This paper investigates the topological structure induced by quasi-valuations on field extensions and establishes that the quasi-valuation ring uniquely determines this topology, along with a weak approximation theorem for quasi-valuations.

Contribution

It proves that the quasi-valuation ring uniquely determines the topology and establishes a weak approximation theorem for quasi-valuations, extending classical valuation theory.

Findings

The topology induced by a quasi-valuation is determined solely by its valuation ring.

The paper proves a weak approximation theorem for quasi-valuations.

The topology is independent of the specific quasi-valuation chosen, given the valuation ring.

Abstract

Suppose $F$ is a field with a nontrivial valuation $v$ and valuation ring $O_{v}$, $E$ is a finite field extension and $w$ is a quasi-valuation on $E$ extending $v$. We study the topology induced by $w$. We prove that the quasi-valuation ring determines the topology, independent of the choice of its quasi-valuation. Moreover, we prove the weak approximation theorem for quasi-valuations.