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Cyclotomic equation over C/Explicit description/Fact/Proof

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Proof

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

Hence, the given complex numbers are indeed zeroes of the polynomial . These zeroes are all different, because

with implies, by considering the fraction, that

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