The Discriminant of a Cubic

The discriminant of a quadratic equation

        ax² + bx + c = 0

is

        Δ = b² – 4ac.

If the discriminant Δ is zero, the equation has a double root (i.e. there is a unique x that makes the equation zero, and it counts twice as a root). If the discriminant is not zero, there are two distinct roots.

Cubic equations also have a discriminant. For a cubic equation

        ax³ + bx² + cx + d = 0

the discriminant is given by

        Δ = 18abcd – 4b³d + b²c² – 4ac³ – 27a²d².

If Δ = 0, the equation has a multiple root. Otherwise, it has three distinct roots.

A change of variable can reduce the general cubic equation to a so-called “depressed” cubic equation of the form

        x³ + px + q = 0

in which case the discriminant simplifies to

        Δ = – 4p³ – 27q².

Here are a couple of interesting connections. The idea of reducing a cubic equation to a depressed cubic goes back to Cardano (1501–1576). What’s called a depressed cubic in this context is known as the Weierstrass (1815–1897) form in the context of elliptic curves. That is, an elliptic curve of the form

        y² = x³ + ax + b

is said to be in Weierstrass form. In other words, an elliptic curve is in Weierstrass form if the right-hand side is a depressed cubic.

Furthermore, an elliptic curve is required to be non-singular, which means it must satisfy

        4a³ + 27b² ≠ 0.

In other words, the discriminant of the right-hand side is non-zero. In the context of elliptic curves, the discriminant is defined to be

        Δ = -16(4a³ + 27b²)

which is the same as the discriminant above, except for a factor of 16 that simplifies some calculations with elliptic curves.

A Note on Fields

In the context of solving quadratic and cubic equations, we’re usually implicitly working with real or complex numbers. Suppose all the coefficients of a quadratic equation are real. If the discriminant is positive, there are two distinct real roots. If the discriminant is negative, there are two distinct complex roots, and these roots are complex conjugates of each other.

Similar remarks hold for cubic equations when the coefficients are all real. If the discriminant is positive, there are three distinct real roots. If the discriminant is negative, there is one real root and a complex conjugate pair of complex roots.

In the first section I only considered whether the discriminant was zero, and so the statements are independent of the field the coefficients come from.

For elliptic curves, one works over a variety of fields. Maybe real or complex numbers, but also finite fields. In most of the blog posts I’ve written about elliptic curves, the field is integers modulo a large prime.