ON EISENSTEIN-DUMAS AND GENERALIZED SCHONEMANN 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 v(ai)/n-i >= v(a(0))/n > 0 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.