Extensions of degree 5, cubic hypersurfaces and covariants (following D. Coray and H. Kraft)

We are interested in the following question: given the separable extension of degree 5 (with some restrictions on the charasteristic of the ground field), how do we find a generating element whose equation is as simple as possible? We will consider two approaches: the first one, which is close to original argument of Hermite, is to construct suitable homogenous $S_5$-covariant, while the second one is based on rationality questions for some cubic surface associated with the field. The latter method includes study of the lines on the surface and Galois group action on them.