Difference between revisions of "Ferrari method"
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| − | + | {{MSC|12Exx}} | |
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| + | The <i> Ferrari method </i> 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_of_a_field|characteristic]] $\ne 2,3$) to the | ||
solution of one cubic and two quadratic equations; it was discovered | solution of one cubic and two quadratic equations; it was discovered | ||
by L. Ferrari (published in 1545). | by L. Ferrari (published in 1545). | ||
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$$=(x^2+\frac{p}{2}+\a)^2-\big[2\a x^2 - qx +\big(\a^2+p\a+\frac{p^2}{4}-r\big)\big].\label{2}$$ | $$=(x^2+\frac{p}{2}+\a)^2-\big[2\a x^2 - qx +\big(\a^2+p\a+\frac{p^2}{4}-r\big)\big].\label{2}$$ | ||
| − | One then chooses a value of $\a$ such that the | + | One then chooses a value of $\a$ such that the quadratic [[Polynomial|trinomial]] in |
| − | square brackets is a perfect square. For this the discriminant of the | + | square brackets is a perfect square. For this the [[Quadratic equation|discriminant of the |
| − | quadratic trinomial must vanish. This gives a | + | quadratic trinomial]] must vanish. This gives a |
[[Cubic equation|cubic equation]] for $\a$, | [[Cubic equation|cubic equation]] for $\a$, | ||
| − | $$q^2- | + | $$q^2-4\cdot2\a\big(\a^2+p\a+\frac{p^2}{4}-r\big)=0.$$ |
Let $\a_0$ be one of the | Let $\a_0$ be one of the | ||
roots of this equation. For $\a=\a_0$ the polynomial in square brackets in | roots of this equation. For $\a=\a_0$ the polynomial in square brackets in | ||
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$$x_0 = \frac{q}{4\a_0},$$ | $$x_0 = \frac{q}{4\a_0},$$ | ||
which leads to the equation | which leads to the equation | ||
| − | $$(x^2+\frac{p}{2}+\a_0)^2 - 2\a_0(x-x_0)^2 = 0.$$ | + | $$\big(x^2+\frac{p}{2}+\a_0\big)^2 - 2\a_0(x-x_0)^2 = 0.$$ |
This | This | ||
| − | equation of degree 4 splits into two quadratic equations. The roots of | + | equation of degree 4 splits into two [[Quadratic equation|quadratic equations]]. The roots of |
these equations are also the roots of (1). | these equations are also the roots of (1). | ||
====References==== | ====References==== | ||
| − | + | {| | |
| − | + | |- | |
| − | + | |valign="top"|{{Ref|Ku}}||valign="top"| A.G. Kurosh, "Higher algebra", MIR (1972) (Translated from Russian) {{ZBL|0237.13001}} | |
| + | |||
| + | |- | ||
| + | |} | ||
Latest revision as of 13:05, 17 December 2014
2020 Mathematics Subject Classification: Primary: 12Exx [MSN][ZBL]
The Ferrari method is a method for reducing the solution of an equation of degree 4 over the complex numbers (or, more generally, over any field of characteristic $\ne 2,3$) to the solution of one cubic and two quadratic equations; it was discovered by L. Ferrari (published in 1545).
The Ferrari method for the equation $$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 |
Ferrari method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ferrari_method&oldid=19674