\begin{definition}[Definition:Discriminant of Polynomial]
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:
:$\ds \map \Delta f := \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {\alpha_i - \alpha_j}^2$
\end{definition}