On Eisenstein–Dumas and Generalized Schönemann Polynomials

Let v be a valuation of a field K having value group ℤ. It is known that a polynomial x n + a n−1 x n−1 + … +a 0 satisfying with v(a 0) 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 Schonemann polynomials with coefficients over valued fields of arbitrary rank.