Cyclotomic equation over C/Explicit description/Fact/Proof

Proof

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

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

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

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

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

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

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

To fact