\prod_{k=1}^{n-1} \exp(i k\pi/n) = \exp \left( \sum_{k=1}^{n-1} {i k\pi \over n} \right) = \exp \left( {i \pi \over n} \sum_{k=1}^{n-1} k \right) = \exp \left( {i \pi \over n} {n(n-1) \over 2} \right) = \exp \left( {i \pi \over 2}(n-1) \right)