Cyclotomic equation over C/Explicit description/Fact

Let ${}n\in \mathbb {N} _{+}$.

The zeroes of the polynomials ${}X^{n}-1$ over ${}\mathbb {C}$ are

$e^{2\pi {\mathrm {i} }k/n}=\cos {\frac {2\pi k}{n}}+{\mathrm {i} }\sin {\frac {2\pi k}{n}},k=0,1,\ldots ,n-1.$

In ${}\mathbb {C} [X]$, we have the factorization

${}X^{n}-1=(X-1){\left(X-e^{2\pi {\mathrm {i} }/n}\right)}\cdots {\left(X-e^{2\pi {\mathrm {i} }(n-1)/n}\right)}\,.$

Proof, Write another proof