A generalized Dumas irreducibility criterion

TL;DR

This paper extends Dumas's irreducibility criterion to polynomials over any valued field with a Krull valuation, providing a unified framework and sharp bounds on factor degrees.

Contribution

It introduces a generalized irreducibility criterion and degree bounds for polynomials over valued fields with Krull valuations, unifying and extending previous results.

Findings

Proved a factorization result for polynomials over valued fields with Krull valuations.

Established lower degree bounds on factors of polynomials.

Unified several known irreducibility results within a broader framework.

Abstract

As an extension of the classical irreducibility result of Dumas, a factorization result for polynomials over any valued field with a Krull valuation of arbitrary rank is proved. Further, a lower degree factor bound on factors of a given polynomial over a valued field with a Krull valuation is proved. These factorization results not only unify several known irreducibility results for polynomials over the said domains but also provide us sharp bounds on degrees of irreducible factors of the underlying polynomials.