A generalization of Eisenstein–Schönemann irreducibility criterion

Abstract

One of the results generalizing Eisenstein Irreducibility Criterion states that if $${\phi(x) = a_nx^n\,{+} \,a_{n-1}x^{n-1} \,{+} \,\cdots\,{+} \,a_0}$$ is a polynomial with coefficients from the ring of integers such that a s is not divisible by a prime p for some $${s \, \leqslant \, n}$$ , each a i is divisible by p for $${0 \, \leqslant \, i \, \leqslant \, s-1}$$ and a 0 is not divisible by p 2, then $${\phi(x)}$$ has an irreducible factor of degree at least s over the field of rational numbers. We have observed that if $${\phi(x)}$$ is as above, then it has an irreducible factor g(x) of degree s over the ring of p-adic integers such that g(x) is an Eisenstein polynomial with respect to p. In this paper, we prove an analogue of the above result for a wider class of polynomials which will extend the classical Schonemann Irreducibility Criterion as well as Generalized Schonemann Irreducibility Criterion and yields irreducibility criteria by Akira et al. (J Number Theory 25:107–111, 1987).