I have been looking into the formulae for the cubic & quartic polynomials, using the resolvent method to come up a corresponding resolvent polynomial of a lesser degree. It seems that it for the case of the given cubic & quartic, the discriminant for the given polynomial is equal to some scale (i.e., my definition here of "congruent") of the discriminant for the corresponding resolvent polynomial (there is no such situation for the quadratic as the resolvent polynomial is linear). And of course, since there is no solution to the quintic polynomial, there is no resolvent.
An example of this is the cubic having the solutions $\{ 0 , +3 , -3 \}$, thus with the discriminant of $2916$. The resolvents are a quadratic solution $\{ \pm 27^{1/2}i \}$, and thus with the discriminant of $-108$, which is $-1/27$ of $2916$. This $-1/27$ scale factor exists for all cubic polynomials, and indeed is the reason why for the case of 3 real solutions, an "intermediate" imaginary solution needs to be calculated (i.e., the resolvent is this "intermediate" solution).
Similarly for a quartic, there is a cubic solution for the resolvents, but in this case, the discriminants are the same. In general, it could be said that the discriminants differ only be a scale factor (which could be the degenerate scale factor of 1). Since there is no solution to the quintic, there is no such thing as a resolvent, and so the consideration of such is moot.
I was just wondering if there is some deeper mathematical reason why these discriminants are scale factors of each other, other than simply saying, "it is this way for the only 2 cases in which it exists".