Discriminant of polynomial (reviewed)

The discriminant of a monic polynomial $f(x) = \prod_{i=1}^d (x - \alpha_i)$ is the quantity \[ \Delta = \prod_{i < j} (\alpha_i - \alpha_j)^2. \] If $f$ has integral coefficients, $K$ is the number field defined by $f$ and $\alpha$ is a root of $f$ in $K$, then the discriminant $D$ of $K$ divides $\Delta$ and the ratio $\Delta/D$ is the square of the index of $\Z[\alpha]$ in the ring of integers of $K$.