Definition:Discriminant of Polynomial

Definition
Let $k$ be a field.

Let $f \left({X}\right)\in k \left[{X}\right]$ be a polynomial of degree $n$.

Let $\overline k$ be an algebraic closure of $k$.

Let the roots of $f$ in $\overline k$ be $\alpha_1, \alpha_2, \ldots, \alpha_n$.

Then the discriminant $\Delta \left({f}\right)$ of $f$ is defined as:


 * $\displaystyle \Delta \left({f}\right) := \prod_{1 \mathop \le i \mathop < j \mathop \le n} \left({\alpha_i - \alpha_j}\right)^2$

Quadratic Equation
The concept is usually encountered in the context of a quadratic equation $a x^2 + b x + c$:

Cubic Equation
In the context of a cubic equation $a x^3 + b x^2 + c x + d$:

Also see

 * Definition:Resultant