Product of Differences of nth Roots of Unity

Question

I'm trying to show that

$$\prod_{j=1}^{n-1}\left(1-e^{2\pi ij/n}\right)=n$$

but am finding it surprisingly difficult. I know by symmetry that

$$\prod_{j=1}^{n-1}\left(1-e^{2\pi ij/n}\right)=\prod_{j=1}^{n-1}\left|1-e^{2\pi ij/n}\right|.$$

I can't seem to get much farther. Any references would be appreciated.

Answer 1

Factor $x^n-1$...

$$\displaystyle\prod_{j=1}^{n-1}(\color{Red}1-e^{2\pi ij/n})=\color{Red}{1}^{n-1}+\cdots+\color{Red}{1}^1+\color{Red}{1}^0$$

(Apparently spoilers don't grey out colored $\LaTeX$ so I'm giving it away.)