On Wirsing's problem in small exact degree

We investigate a variant of Wirsing's problem on approximation to a real number by real algebraic numbers of degree exactly $n$. This has been studied by Bugeaud and Teulie. We improve their bounds for degrees up to $n=7$. Moreover, we obtain results regarding small values of polynomials and approximation to a real number by algebraic integers in small prescribed degree. The main ingredient are irreducibility criteria for integral linear combinations of coprime integer polynomials. Moreover, for cubic polynomials these criteria improve results of Gy\H{o}ry on a problem of Szegedy.