Resolvent cubic

In algebra, a resolvent cubic is one of several distinct, although related, cubic polynomials defined from a monic polynomial of degree four:
In each case:
Definitions.
Suppose that the coefficients of belong to a field whose characteristic is different from . In other words, we are working in a field in which . Whenever roots of are mentioned, they belong to some extension of such that factors into linear factors in . If is the field of rational numbers, then can be the field of complex numbers or the field of algebraic numbers.
In some cases, the concept of resolvent cubic is defined only when is a quartic in depressed form—that is, when .
Note that the fourth and fifth definitions below also make sense and that the relationship between these resolvent cubics and are still valid if the characteristic of is equal to .
First definition.
Suppose that is a depressed quartic—that is, that . A possible definition of the resolvent cubic of is:
The origin of this definition lies in applying Ferrari's method to find the roots of . To be more precise:
\left(x^2+\frac{a_2}2\right)^2=-a_1x-a_0+\frac