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

On Wirsing's problem in small exact degree