Cyclotomic equation over C/Explicit description/Fact/Proof

The proof uses some basic facts about the complex exponential function. We have

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

Hence, the given complex numbers are indeed zeroes of the polynomial ${\displaystyle {}X^{n}-1}$. These zeroes are all different, because

${\displaystyle {}e^{2\pi {\mathrm {i} }k/n}=e^{2\pi {\mathrm {i} }\ell /n}\,}$

with ${\displaystyle {}0\leq k\leq \ell \leq n-1}$ implies, by considering the fraction, ${\displaystyle {}e^{2\pi {\mathrm {i} }(\ell -k)/n}=1}$ that

${\displaystyle {}\ell -k=0\,}$

holds. Therefore, there exist ${\displaystyle {}n}$ explicit zeroes and these are all the zeroes of the polynomial. The explicit description in coordinates follows from the Euler's formula.