A generalization of Dumas-Eisenstein criterion

TL;DR

This paper introduces Dumas valuations, a broad class of monoid homomorphisms, and proposes a conjecture that generalizes the Eisenstein-Dumas irreducibility criterion, with implications for polynomial irreducibility over various domains.

Contribution

The paper generalizes the Eisenstein-Dumas criterion by formulating a new conjecture involving Dumas valuations, applicable to multivariate polynomials and algebraic number fields.

Findings

Proposes a new class of valuations called Dumas valuations.

Formulates a conjecture extending classical irreducibility criteria.

Provides evidence supporting the conjecture's validity.

Abstract

We introduce an interesting and rather large class of monoid homomorphisms, on arbitrary integral domain $R$, that we call Dumas valuations. Then we formulate a conjecture addressing the question asking when a polynomial $f\in R[X]$ cannot be written as a product $f=gh$ for some nonconstant polynomials $g,h\in R[X]$. The statement of the conjecture presents a significant generalization of the classical Eisenstein-Dumas irreducibility criterion. In particular our approach can be very useful while studying the irreducibility problem for multivariate polynomials over any integral domain and polynomials over orders in algebraic number fields. We provide a strong evidence that our conjecture should be true.