We study polynomials with integer coefficients which become Eisenstein
polynomials after the additive shift of a variable. We call such polynomials
shifted Eisenstein polynomials. We determine an upper bound on the maximum
shift that is needed given a shifted Eisenstein polynomial and also provide a
lower bound on the density of shifted Eisenstein polynomials, which is strictly
greater than the density of classical Eisenstein polynomials. We also show that
the number of irreducible degree n polynomials that are not shifted
Eisenstein polynomials is infinite. We conclude with some numerical results on
the densities of shifted Eisenstein polynomials