Cyclotomic equation over C/Explicit description/Fact

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

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

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

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

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