A new proof of Rédei’s theorem on the number of directions

Rédei and Megyesi proved that the number of directions determined by a p-element subset of \({\mathbb F}_p^2\) ${\mathbb{F}}_p^2$ is either 1 or at least \(\frac{p+3}{2}\) $\frac{p+3}{2}$ . The same result was independently obtained by Dress, Klin, and Muzychuk. We give a new and short proof of this result using a lemma proved by Kiss and the author. The new proof relies on a result on polynomials over finite fields.