Ferrari method

2020 Mathematics Subject Classification: Primary: 12Exx [MSN][ZBL]

The Ferrari method is a method for reducing the solution of an equation of degree 4 over the complex numbers (or, more generally, over any field of characteristic $\ne 2,3$) to the solution of one cubic and two quadratic equations; it was discovered by L. Ferrari (published in 1545).

The Ferrari method for the equation $$y^4 + a y^3 + by^2 + cy + d = 0$$ consists in the following. By the substitution $y=x-a/4$ the given equation can be reduced to $$x^4+px^2 + qx +r = 0,\label{1}$$ which contains no term in $x^3$. If one introduces an auxiliary parameter $\def\a{\alpha}\a$, the left-hand side of (1) can be written as $$x^4+px^2+qx+r =$$

$$=(x^2+\frac{p}{2}+\a)^2-\big[2\a x^2 - qx +\big(\a^2+p\a+\frac{p^2}{4}-r\big)\big].\label{2}$$ One then chooses a value of $\a$ such that the quadratic trinomial in square brackets is a perfect square. For this the discriminant of the quadratic trinomial must vanish. This gives a cubic equation for $\a$, $$q^2-4\cdot2\a\big(\a^2+p\a+\frac{p^2}{4}-r\big)=0.$$ Let $\a_0$ be one of the roots of this equation. For $\a=\a_0$ the polynomial in square brackets in (2) has one double root, $$x_0 = \frac{q}{4\a_0},$$ which leads to the equation $$\big(x^2+\frac{p}{2}+\a_0\big)^2 - 2\a_0(x-x_0)^2 = 0.$$ This equation of degree 4 splits into two quadratic equations. The roots of these equations are also the roots of (1).

References

[Ku]	A.G. Kurosh, "Higher algebra", MIR (1972) (Translated from Russian) Zbl 0237.13001