On note \(\omega = e^{2i\pi/n}\), \(\alpha = e^{i\pi/n}\) et \(D\) le déterminant \(n \times n\) : \(D = \det\Bigl( \omega^{(k-1)(l-1)} \Bigr)\).
1
Calculer \(D^2\).
2
Montrer que \(D = \prod_{k < \ell}(\omega^\ell - \omega^k) = \prod_{k < \ell}\left(\alpha^{k+\ell}\cdot 2i\sin\frac {\ell-k}n \pi\right)\).
3
Exprimer \(D\) sous forme trigonométrique.