A result of Hermite and equations of degree 5 and 6

A classical result from 1861 due to Hermite says that every separable equation of degree 5 can be transformed into an equation of the form x5+bx3+cx+d=0. Later, in 1867, this was generalized to equations of degree 6 by Joubert. We show that both results can be understood as an explicit analysis of certain covariants of the symmetric groups S5 and S6. In case of degree 5, the classical invariant theory of binary forms of degree 5 comes into play whereas in degree 6 the existence of an outer automorphism of S6 plays an essential rôle.