Cardano’s Formula and Casus Irreducibilis – Dokuz Eylül University Faculty of Science Department of Mathematics

Çağdaş Çiğdemoğlu, Dokuz Eylül University. Date: 18th of December, 2024, Wednesday, Time: 15.00 – 16.00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Classroom B255.

Abstract: We will start with a general monic cubic equation in the form

$x^3 + bx^2 + cx + d = 0,$

and transform it into the following form

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

using a substitution. Then, we will construct the Cardano Formulas to find the roots of the equation. The roots are expressed as follows:

$y_1 = \sqrt[3]{\frac{-q + \sqrt{q^2 + \frac{4p^3}{27}}}{2}} + \sqrt[3]{\frac{-q - \sqrt{q^2 + \frac{4p^3}{27}}}{2}},$

$y_2 = \omega \sqrt[3]{\frac{-q + \sqrt{q^2 + \frac{4p^3}{27}}}{2}} + \omega^2 \sqrt[3]{\frac{-q - \sqrt{q^2 + \frac{4p^3}{27}}}{2}},$

$y_3 = \omega^2 \sqrt[3]{\frac{-q + \sqrt{q^2 + \frac{4p^3}{27}}}{2}} + \omega \sqrt[3]{\frac{-q - \sqrt{q^2 + \frac{4p^3}{27}}}{2}},$

where $\omega = e^{i \frac{2\pi}{3}} = \cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right) = -\frac{1}{2} + i\frac{\sqrt{3}}{2}$ is a primitive cube root of unity and where the above cube roots are one of the three complex cube roots whose product is $-\frac{p}{3}$ and these are fixed in the above formulas.

We will learn about the discriminant $\Delta$ of the above monic cubic polynomial, understand its significance, and examine how the roots change depending on the value of the discriminant.

For the monic cubic polynomial in the form $y^3 + py + q$ , the discriminant is expressed by:

$\Delta = -27q^2 - 4p^3 = (y_1-y_2)^2(y_1-y_3)^2(y_2-y_3)^2.$

For the general monic cubic polynomial $x^3 + bx^2 + cx + d$ , whose roots are $x_1, x_2, x_3$ , the discriminant is expressed by:

$\Delta = b^2c^2 + 18bcd - 4c^3 - 4b^3d - 27d^2 = (x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2.$

We will also discuss Casus Irreducibilis, which occurs when the coefficients $b, c, d$ (or $p, q$ ) are in a subfield of $F$ of $\mathbb{R}$ , the cubic polynomial is irreducible over $F$ (equivalently, the cubic equation has no roots in $F$ ) and when the discriminant is positive. In this case, the cubic equation has three distinct real roots that cannot be expressed using real radicals.

This seminar serves as an introduction to my project, which focuses on understanding the proof of Casus Irreducibilis using Galois Theory.

Cardano's Formula and Casus Irreducibilis