\begin{definition}[Definition:Cubic Equation/Resolvent]
Let $P$ be the cubic equation:
: $a x^3 + b x^2 + c x + d = 0$ with $a \ne 0$
Let:
: $y = x + \dfrac b {3 a}$
: $Q = \dfrac {3 a c - b^2} {9 a^2}$
: $R = \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}$
Let $y = u + v$ where $u v = -Q$.
The '''resolvent equation''' of the cubic is given by:
: $u^6 - 2 R u^3 - Q^3$
\end{definition}