PlanetMath: quadratic resolvent

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quadratic resolvent

(Definition)

The quadratic resolvent of the cubic equation

$\displaystyle y^3+py+q = 0,$

(1)

where

and

are known complex numbers, is the auxiliary equation

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

determining as its roots

$\displaystyle z_1 = u^3, \quad z_2 = v^3$

and

whose sum

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

"quadratic resolvent" is owned by pahio.

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Other names:

quadratic resolvent equation

Also defines:

cubic resolvent, resolvent equation

Keywords:

resolvent

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Cross-references: resolvent cubic, quartic equation, example of solving a cubic equation, sum, numbers, equation, complex numbers, cubic equation
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This is version 4 of quadratic resolvent, born on 2008-02-26, modified 2008-02-27.
Object id is 10338, canonical name is QuadraticResolvent.
Accessed 571 times total.

Classification:

12D10 (Field theory and polynomials :: Real and complex fields :: Polynomials: location of zeros )

Pending Errata and Addenda

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