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A generalization of Dumas irreducibility criterion

Jitender Singh

TL;DR

The paper generalizes Dumas' irreducibility criterion by leveraging Newton polygons to obtain new tests for irreducibility of polynomials over discrete valuation domains. It develops a broad factorization framework: a mild generalization that enforces a lower bound on the degree of any factor, plus p-adic and size-based corollaries for integer-coefficient polynomials, including zeros lying outside the unit disk. The results are shown to be sharp via explicit constructions and extend to a multi-prime setting yielding a bound on the number of irreducible factors. Overall, the work broadens the Newton polygon method as a practical irreducibility tool.

Abstract

Using Newton polygons, a key factorization result for polynomials over discrete valuation domains is proved, which in particular yields new irreducibility criteria including a generalization of the classical irreducibility criterion of Dumas.

A generalization of Dumas irreducibility criterion

TL;DR

The paper generalizes Dumas' irreducibility criterion by leveraging Newton polygons to obtain new tests for irreducibility of polynomials over discrete valuation domains. It develops a broad factorization framework: a mild generalization that enforces a lower bound on the degree of any factor, plus p-adic and size-based corollaries for integer-coefficient polynomials, including zeros lying outside the unit disk. The results are shown to be sharp via explicit constructions and extend to a multi-prime setting yielding a bound on the number of irreducible factors. Overall, the work broadens the Newton polygon method as a practical irreducibility tool.

Abstract

Using Newton polygons, a key factorization result for polynomials over discrete valuation domains is proved, which in particular yields new irreducibility criteria including a generalization of the classical irreducibility criterion of Dumas.
Paper Structure (3 sections, 9 theorems, 30 equations)

This paper contains 3 sections, 9 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proofs

Key Result

Theorem 1

Let $(R,v)$ be a discrete valuation domain. Let $f=a_0 + a_1x + \cdots + a_nx^n \in {R}[x]$ be a primitive polynomial such that there exist indices $j$ and $\ell$ with $1\leq \ell+1 \leq j\leq n$ for which the following hold. Then any factorization $f(x)=f_1(x)f_2(x)$ of $f$ in $R[x]$ has a factor of degree $\geq j-\ell$. In particular, if $j=n$ and $\ell=0$, then the polynomial $f$ is irreducibl

Theorems & Definitions (17)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Corollary 4
  • Corollary 5
  • Theorem 6
  • Lemma 7
  • proof : Proof of Lemma \ref{['L:1']}
  • proof : Proof of Theorem \ref{['th:1']}
  • proof : Proof of Corollary \ref{['c:1']}
  • ...and 7 more