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.

Key Words:

• Irreducible polynomials
• Newton polygon
• Prime numbers

2010 Mathematics Subject Classification:

• Primary: 11R09
• Secondary: 11C08

Notes

Communicated by T. Albu.