The number of complex numbers $z$ such that $|z|=1$ and $|z^4+z^3+z^2+z+1|=0$ is

Solution

1. The given equation $|z^4+z^3+z^2+z+1|=0$ implies $z^4+z^3+z^2+z+1=0$.
2. Using the formula for a geometric series, this is equivalent to $\frac{z^5-1}{z-1}=0$ for $z\ne 1$.
3. The roots of $z^5-1=0$ are the five fifth roots of unity, given by $z=e^{i\frac{2k\pi }{5}}$ for $k=0,1,2,3,4$.
4. Excluding $z=1$ (which corresponds to $k=0$), there are exactly $4$ distinct complex numbers that satisfy the equation.
5. Hence the answer is (C).