The dielectric function (DF) of hexagonal AlN on Si(111) is determined in the range between 1 and 9.8 eV by spectroscopic ellipsometry (SE). Due to its large negative crytal-field splitting wurtzite AlN features large dichroism. Showing that SE is sensitive to both components of the DF around the absorption edge, a uniaxial model is applied which yields transition energies for the free excitonic state. The in-plane tensile stress leads to a red-shift of these transitions and to an enlarged splitting. The experimental data are compared to the results of band-structure calculations demonstrating excellent overall agreement. In addition, two high-energy critical points in the ordinary DF were determined at energies of about 7.75 and 8.85 eV.