Schönemann–Eisenstein–Dumas-Type Irreducibility Conditions that Use Arbitrarily Many Prime Numbers

Abstract

The famous irreducibility criteria of Schönemann-Eisenstein and Dumas rely on information on the divisibility of the coefficients of a polynomial by a single prime number. In this paper, we will use some results and ideas of Dumas to provide several irreducibility criteria of Schönemann-Eisenstein-Dumas-type for polynomials with integer coefficients, criteria that are given by some divisibility conditions for their coefficients with respect to arbitrarily many prime numbers. A special attention will be paid to those irreducibility criteria that require information on the divisibility of the coefficients by two distinct prime numbers.

On Polynomial Modular Number Systems over $\mathbb{Z}/p\mathbb{Z}$

“…Proof. It is a direct application of the Corollary 1.2 of a paper due to Nicolae Ciprian Bonciocat [6].…”

“…To verify the first item, we can use general criteria like Schönemann-Eisenstein or Dumas [8] or the generalization given by N. C. Bonciocat in [6], that we adapt to our purpose : a monic polynomial E…”

On the Factorization of Polynomials Over Discrete Valuation Domains

“…There are many recent results that provide irreducibility conditions for various classes of polynomials by using techniques coming from valuation theory (see for instance [23], [24], [2], [3], [4], [8], [5] and [9]), or Newton polygon method (see for instance [12], [13], [14], [15], [16], [17], [18], [1], [6], [7], [22] and [25]). …”