Sch\"onemann-Eisenstein-Dumas-type irreducibility conditions that use arbitrarily many prime numbers

TL;DR

This paper extends classical irreducibility criteria to include conditions involving multiple prime numbers, providing new tools for determining polynomial irreducibility with richer divisibility information.

Contribution

It introduces novel irreducibility criteria based on divisibility conditions involving arbitrarily many primes, especially focusing on two primes, expanding classical methods.

Findings

New irreducibility criteria involving multiple primes

Criteria that use divisibility by two distinct primes

Enhanced methods for polynomial irreducibility testing

Abstract

The famous irreducibility criteria of Sch\"onemann-Eisenstein and Dumas rely on information on the divisibility of the coefficients of a polynomial by a single prime number. In this paper we provide several irreducibility criteria of Sch\"onemann-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.