quadratic resolvent

The quadratic resolvent of the cubic equation

$y^3+py+q=0,$ (1)

where $p$ and $q$ are known complex numbers, is the auxiliary equation

$$z^2+qz-{\left(\frac{p}{3}\right)}^3=0$$

determining as its roots (http://planetmath.org/Equation)

$$z_1=u^3,z_2=v^3$$

the numbers $u$ and $v$ whose sum  $y=u+v$  satisfies the equation (1).  See example of solving a cubic equation (http://planetmath.org/exampleofsolvingacubicequation).

Analogically, a quartic equation has a cubic resolvent (resolvent cubic) equation.

Title	quadratic resolvent
Canonical name	QuadraticResolvent
Date of creation	2014-11-27 16:05:30
Last modified on	2014-11-27 16:05:30
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	9
Author	pahio (2872)
Entry type	Definition
Classification	msc 12D10
Synonym	quadratic resolvent equation
Related topic	CardanosFormulae
Related topic	TchirnhausTransformations
Defines	cubic resolvent
Defines	resolvent equation