A Generalization of Eisenstein-Schönemann’s Irreducibility Criterion

Abstract

The Eisenstein criterion is a particular case of the Schönemann’s irreducibility criterion stated in 1846. In 1906, based on Newton polygon techniques, Dumas gave a generalization of the Eisenstein criterion. In this paper, we extend this last generalization. Some applications on factorization of polynomials, and prime ideal factorization will be given, too.

On Indices of Septic Number Fields Defined by Trinomials x7 + ax + b

“…Then, N φ (F) = S has a single side of degree 1. Thus by Remark 2 (2) of [22], F(x) is irreducible over Q. Let θ be a root of F(x).…”