In this paper, we show that for any monic nonconstant polynomial with complex coefficients P and any \(\varepsilon >0\) there is a monic irreducible polynomial Q with integer coefficients such that the modulus of the resultant of P and Q is less than \(\varepsilon \) . In particular, for any complex number z, for \(P(x)=x-z \in {\mathbb {C}}[x]\) and \(\varepsilon =1\) , this implies that there exists a monic irreducible polynomial \(Q \in {\mathbb {Z}}[x]\) for which \(|Q(z)|<1\) . In this context, we are also interested in the smallest possible degree \(D_z\) of such a polynomial Q. We prove that \(D_z \le 4\) for all \(z \in {\mathbb {C}}\) , except for some quadratic algebraic numbers z. However, there is no universal upper bound on \(D_z\) . We show that if \(\ell \ge 2\) and \(k>\ell ^{2m-2}\) are positive integers, then the inequality \(D_z > m\) holds for z satisfying \(z^2-\ell ^{-1}z+k=0\) .
