Let us consider, for rational numbers a,b,c,d, an equation
Let
be the roots of this equation and let L be a field generated by the set Q of rational numbers and these roots.
The Galois group Gal L/Q of this equation is a group S4 of all permutation of four elements. We assume that these elements are roots
x1, x2, x3, x4.
The discriminant of this equation is a number s defined by
The expression is invariant under acting of the group Gal L/K, hence it must be rational. In fact, we can express s in terms of coefficients of the
equation. Number s (discriminant) is equal
Dropping powers in a formula defining s, we can easily verify that
Let us define
Then
and
We can verify that
Applying formulas for symmetric functions and formulas of Vieta, we see that
All last equalities, they are formulas of Vieta of the equation
It follows that A1, A2, A3 are the roots of this equation. Formulas of Cardano for roots of cubic equation give us
where w is a root of 1 of third degree defined by
and u is a number equal
From the definition of the numbers A1, A2, A3 and a simple computation it follows that
Taking square roots, adding one equality from formulas of Vieta, we receive a system of four linear equations with four unknowns
Solving this system, we receive final formulas
Click here to see a similar page of Literka for a solution of
cubic equation.
If you are interested in mathematical software about cubic equations click new module "Cubic Functions"
of a free program "Ruler and Compass" or download this program.
There is something like an eBook about quartic and cubic equations. See
"Mathematical Reader" to learn more.