How to find a complex root of $x^{2021}=x^{2020}+1$ while this complex root also satisfies a quadratic equation with integer coefficients? How to find a complex root of $x^{2021}=x^{2020}+1$ while this complex root also satisfies a quadratic equation with integer coefficients? I have no previous experience in solving complex equations so just have no clue on this kind of question... Is there any useful pattern on this?

Hint: $x^5 - x^4 - 1 = (x^2 - x + 1) ( x^3 - x - 1)$.   
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 Hint: $x^3 +1 = (x^2 - x + 1 ) ( x + 1)$.

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 Hence, $ x^{6n+k} \equiv x^{k} \pmod{x^2-x+1}$

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 Hence $ x^{2021} - x^{2020} - 1 \equiv x^5 - x^4 - 1 \equiv 0 \pmod{x^2-x+1}$.

