About Eisenstein's Theorem

TL;DR

This survey paper discusses Eisenstein's theorem, provides a generalized version, and explores applications related to the algebraic closure of multivariate power series fields.

Contribution

It introduces a generalized form of Eisenstein's theorem and demonstrates its applications in algebraic closure studies of power series fields.

Findings

Generalized Eisenstein's theorem proved.

Applications to algebraic closure of power series fields.

Enhanced understanding of power series algebraic structures.

Abstract

The aim of this survey papier is to present a result due to Eisenstein, to prove a generalized version of it, and to present some applications of this Eisenstein's Theorem, in particular to the study of the algebraic closure of the field of power series in several indeterminates.