Precalculus please help

Question

What is the smallest positive integer $n$ such that all the roots of $z^4 + z^2 + 1 = 0$ are $n$th roots of unity?

Answer 1

Interesting query re pre-calculus tag combined with a complex analysis problem.

Hint:

The $n$-th roots of unity are the roots of

$$0 = (z^n - 1) = (z - 1)\left[z^{(n-1)} + z^{(n-2)} + \cdots + z^2 + z + 1\right]. \tag1$$

Therefore, one approach to the challenge is to look for a way to convert

$$\left[z^4 + z^2 + 1\right]$$

into the rightmost factor of equation (1) above.

One obvious try is to look for a specific value of $n$ such that $$z^4 = z^1 \implies \{z^3 = 1 ~\text{or}~ z = 0\}.$$