IME

Practice session

Let $z \in \mathbb{C}$ be a root of $z^5 + z^4 + z^3 + z^2 + z + 1 = 0$ with $\operatorname{Im}(z) > 0$. What is the value of $\operatorname{Re}(z) + \operatorname{Re}(z^2) + \operatorname{Re}(z^3) + \operatorname{Re}(z^4) + \operatorname{Re}(z^5)$?

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