Definition:Discriminant

Definition

Let $f \left({x}\right)$ be a polynomial in $x$ of degree $n$ over a field $k$.

That is, let $f \left({x}\right) = x^n + a_1 x^{n-1} + a_2 x^{n-2} + \cdots + a_{n-1} x + a_n$ with $a_i \in k$, $i = 1,\ldots, n$.

Let the roots of the equation $f \left({x}\right) = 0$ be $\alpha_1, \alpha_2, \ldots, \alpha_n$.

Then the discriminant $\Delta(f)$ of $f$ is defined as:

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