In Higher composition laws IV: The parametrization of quintic rings M. Bhargava gave an explicit parametrization of quintic rings by quadruples of $5\times5$ skew-symmetric matrices. His proof hinges on establishing a previously unknown fundamental resolvent map between the triple product of the resolvent ring $S$ and the dual of the quintic ring $R$. The mere existence of this map is astonishing in and of itself.

My question is, what can we say about sextic rings which are sextic resolvent rings of some quintic ring? In particular, is every sextic ring a resolvent ring for some quintic ring?

For lower ranks, the answer is yes: every cubic ring is the cubic resolvent ring of some quartic ring, and every quadratic ring is the quadratic resolvent ring of some cubic ring.

In Higher composition laws IV: The parametrization of quintic rings M. Bhargava gave an explicit parametrization of quintic rings by quadruples of $5\times5$ skew-symmetric matrices. His proof hinges on establishing a previously unknown fundamental resolvent map between the triple product of the resolvent ring $S$ and the dual of the quintic ring $R$. The mere existence of this map is astonishing in and of itself.

My question is, what can we say about sextic rings which are sextic resolvent rings of some quintic ring?

We can see that not all sextic rings can be sextic resolvent rings of quintics, since the discriminant relation

$$\displaystyle \operatorname{Disc}(S) = (16 \cdot \operatorname{Disc}(R))^3$$

holds. In particular, every sextic resolvent ring has discriminant equal to a cube.