## Compute the discriminant of a given field extension

Compute the discriminant of $\mathbb{Q}(\alpha)$, where $\alpha$ is a root of $p(x) = x^2 + 5x + 17$.

Evidently, the roots of $p(x)$ are $(-5 \pm \sqrt{-43})/2$; let $\alpha$ be the root with the positive option taken. The discriminant of $\mathbb{Q}(\alpha)$ is then $\mathsf{det}^2\ \left[ \begin{array}{cc} 1 & (-5+\sqrt{-43})/2 \\ 1 & (-5-\sqrt{-43})/2 \end{array} \right] = -43$.