Pentagram

It is given that $f(z) = z^4+z^3+z^2+z+1 = 0$ .

Multiplying it with $z$ : $\Rightarrow z^5+ z^4+z^3+z^2+z= 0$

Adding $1$ on both sides: $\Rightarrow z^5+ z^4+z^3+z^2+z + 1= 1$

$\Rightarrow z_1^5 = z_2^5 = z_3^5 = z_4^5 = 1\quad \Rightarrow \displaystyle \sum_{i=1}^4 {z_i^5} = \boxed{4}$

The given equation is $\frac {z^5 -1}{z-1} = 0$ which means that $z_1 , z_2 , z_3 , z_4$ are four out of the five fifth roots of unity except 1.

$z_1^5 + z_2^5 + z_3^5 + z_4^5 + 1^5 = 5$

$\displaystyle\sum_{i=1}^4 z_i ^5 = 4$

They are complex fifth roots of unity.

So, $z_1^5 + z_2^5 + z_3^5 + z_4^5 + 1^5 = 5$

$\displaystyle\sum_{i=1}^4 z_i ^5 = 4$