Multivariate Hensel Lemma for ultrametric fields

TL;DR

This paper provides a direct, constructive proof of the Multivariate Hensel Lemma specifically for ultrametric fields, avoiding reliance on the Grothendieck Zariski Main Theorem, and extends to rank-one valued fields in classical mathematics.

Contribution

It offers a novel, constructive proof of the Multivariate Hensel Lemma for ultrametric fields, simplifying the proof process and broadening its applicability.

Findings

Direct proof of the lemma for ultrametric fields

Extension to rank-one valued fields in classical mathematics

Avoidance of Zariski Main Theorem in proof

Abstract

The Multivariate Hensel Lemma for local rings is usually proved as a consequence of the Grothendieck version of Zariski's Main Theorem. This version deals with a more general situation that is a priori much more difficult. In this paper, we give a direct proof of the Multivariate Hensel Lemma for ultrametric fields, in the framework of constructive mathematics and without using~ZMT. In the framework of classical mathematics, our result entails the Lemma for rank-one valued fields.