On Eisenstein–Dumas and generalized Schönemann polynomials

Abstract

Let v be a valuation of a field K having value group Z. It is known that a polynomial xⁿ + a_n−1xⁿ⁻¹ + … +a₀ satisfying v(a_i)/n-i≥v(a₀) > 0 with v(a₀) coprime to n, is irreducible over K. Such a polynomial is referred to as an Eisenstein–Dumas polynomial with respect to v. In this article, we give necessary and sufficient conditions so that some translate g(x + a) of a given polynomial g(x) belonging to K[x] is an Eisenstein–Dumas polynomial with respect to v. In fact, an analogous problem is dealt with for a wider class of polynomials, viz. Generalized Schönemann polynomials with coefficients over valued fields of arbitrary rank.