A result of Hermite and equations of degree 5 and 6

classical result from 1861 due to HERMITE says that every separable equation of degree 5 can be transformed into an equation of the form x(5) + bx(3) + 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 S-5 and S-6. 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 S-6 plays an essential role. (c) 2005 Elsevier Inc. All rights reserved.