A generalized Dumas irreducibility criterion

TL;DR

The paper generalizes the Dumas irreducibility criterion to polynomials over fields endowed with a Krull valuation of arbitrary rank, yielding sharp lower bounds on the degrees of irreducible factors. It extends Newton-polygon–style methods to rank-valued settings by constructing valuations $w_j$ on $\mathbb{K}[z]$ and using a combined value group $G=\langle G_v,\gamma_j\rangle$, where $\gamma_j=v(a_k)/(j-k)$. Key results include Theorem 1 and Theorem 2, which unify several prior irreducibility criteria (e.g., JK, JRG2, Weintraub) and provide concrete degree bounds such as $\text{deg}(f_i)\le n-j+k$ (or $\delta_f$ in general). These contributions broaden irreducibility testing and factorization bounds over general valued fields, with implications for algebraic number theory and computational applications.

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.

Theorems & Definitions (6)

• Theorem 1
• Corollary 2
• Theorem 3
• proof : Proof of Theorem \ref{['th:1']}
• proof : Proof of Corollary \ref{['c:1']}
• proof : Proof of Theorem \ref{['th:2']}