Let $\eta=e^{2\pi i/n}$, an $n$-th root of unity. For pedagogical reasons and inspiration, I ask to see different proofs (be it elementary, sophisticated, theoretical, etc) for the following product evaluation.

If $T(n)=\frac{(3n-2)(n-1)}2$ and $i=\sqrt{-1}$ then $$\prod_{j<k}^{0,n-1}(\eta^k-\eta^j)=n^{\frac{n}2}i^{T(n)}.$$

Let $\eta=e^{\frac{2\pi i}n}$, an $n$-th root of unity. For pedagogical reasons and inspiration, I ask to see different proofs (be it elementary, sophisticated, theoretical, etc) for the following product evaluation.

If $T(n)=\frac{(3n-2)(n-1)}2$ and $i=\sqrt{-1}$ then $$\prod_{j<k}^{0,n-1}(\eta^k-\eta^j)=n^{\frac{n}2}i^{T(n)}.$$