Definition:Cubic Equation/Resolvent
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Definition
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:
- $u^6 - 2 R u^3 - Q^3$
Also defined as
Some sources introduce Cardano's Formula starting from the cubic:
- $x^3 + q x - r = 0$
to which the general cubic can be reduced to using the Tschirnhaus Transformation.
In this form, the resolvent equation of the cubic is given by:
- $u^6 - r u^3 - \dfrac q {27}$
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Introduction