Okutsu sequences in Henselian fields

TL;DR

This paper extends the concept of Okutsu sequences to all elements in algebraic closures of Henselian fields, establishing their equivalence to Okutsu frames and Mac Lane-Vaquié chains, thus broadening their applicability.

Contribution

It generalizes Okutsu sequences beyond defectless extensions, linking them to Okutsu frames and Mac Lane-Vaquié chains in Henselian fields.

Findings

Extended Okutsu sequences to arbitrary elements in algebraic closures.

Proved equivalence between Okutsu sequences, frames, and Mac Lane-Vaquié chains.

Unified valuation-theoretic framework for Henselian fields.

Abstract

For $(K,v)$ a Henselian valued field, let $\theta\in\overline{K}$ with minimal polynomial $F$ over $K$. Okutsu sequences of $\theta$ have been defined only when the extension $K(\theta)/K$ is defectless. In this paper, we extend this concept to arbitrary $\theta\in\overline{K}$ and we show that these objects are essentially equivalent to Okutsu frames of $F$ and to Mac Lane-Vaqui\'e chains of the natural valuation on $K[x]$ induced by $\theta$.