Construction du hens\'elis\'e d'un corps valu\'e

TL;DR

This paper provides an explicit, constructive method for obtaining the henselization of a valued field, paralleling the real closure construction for ordered fields, with a focus on mathematical rigor and clarity.

Contribution

It introduces a new explicit, constructive approach to the henselization of valued fields, enhancing understanding and potential applications in valuation theory.

Findings

Constructive proof of henselization for valued fields

Explicit construction method detailed

Analogy with real closure of ordered fields

Abstract

We give an explicit construction of the henselization of a valued field, with a constructive proof. It is analogous to the construction of the real closure of a discrete ordered field. Nous donnons une construction explicite, et constructivement prouv\'ee, du hens\'elis\'e d'un corps valu\'e. Cette construction peut \^etre consid\'er\'ee comme l'analogue, dans le cas valu\'e, de la construction de la cl\^oture r\'e\'eelle d'un corps ordonn\'e.