In a short paper in 1861 Hermite showed that a general equation of degree 5, $$ x^5 + a_1x^4 + a_2 x^3 + a_3x^2 + a_4x + a_5=0, $$ can be reduced to the form $x^5 + ax^3 + bx + b = 0$. Since then, this result and related questions have been studied from different viewpoints, by Felix Klein, David Hilbert, Richard Brauer, Jean-Pierre Serre, Yu I. Manin and others. More recently, Joe Buhler and Zinovy Reichstein found a very interesting connection of these problems with the study of rational covariants of the symmetric group. We will explain this approach and show how it is related to some classical invariant theory. (This is joint work with G.W. Schwarz.)