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.

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