What is the best way to verify that $$1+z+z^2+z^3+z^4=0$$ given $z=e^{2\pi i/5}$?
I tried using Euler's formula before substituting this in, but the work got messy real fast.
Answer
Clearly(?) $z^5=1$ and $z\ne 1$. Also, $$z^ 5-1=(z-1)(1+z+z^2+z^3+z^4)$$
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