# Definition:Discriminant of Polynomial

## Definition

Let $k$ be a field.

Let $\map f X \in k \sqbrk X$ 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 $\map \Delta f$ of $f$ is defined as:

$\displaystyle \map \Delta f := \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {\alpha_i - \alpha_j}^2$

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

The expression $b^2 - 4 a c$ is called the discriminant of the equation.

### Cubic Equation

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

Let:

$Q = \dfrac {3 a c - b^2} {9 a^2}$
$R = \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}$

The discriminant of the cubic equation is given by:

$D := Q^3 + R^2$