A Generalization of the Eisenstein–Dumas–Schönemann Irreducibility Criterion

Abstract

In 2013, Weintraub gave a generalization of the classical Eisenstein irreducibility criterion in an attempt to correct a false claim made by Eisenstein. Using a different approach, we prove Weintraub's result with a weaker hypothesis in a more general setup that leads to an extension of the generalized Schönemann irreducibility criterion for polynomials with coefficients in arbitrary valued fields.

On a factorization result of Ştefănescu

“…Most of the factorization results on polynomials over a discrete valuation domain (R, v) require to take v(a s ) and n − s to be coprime, whenever s is the smallest index for which the minimum of the quantity v(a i )/(n − i), 0 ≤ i ≤ n − 1 is v(a s )/(n − s) (See for example, Jhorar and Khanduja [6]). In the case when v(a s ) and n − s are not coprime, we have the following result.…”

A Useful Irreducibility Result for Integer Polynomials

“…It may be pointed out that, in the above examples, the irreducibility of the polynomials f 5 (x) and g 4 (x) do not seem to follow from any known irreducibility criterion (cf. [1], [2], [3], [4], [5], [6] and [7]).…”