Primitive 16th root of unity
Solution 1:
You can prove this directly from definitions. Obviously $\zeta^2$ is an $8$th root of unity and $\zeta^4$ is a $4$th root of unity, so all you need to show is that they're primitive. Suppose they weren't. That would mean, in the one case, that $\zeta^2$ is a $4$th root of unity, which would mean that $\zeta$ is an $8$th root of unity, which would mean that $\zeta$ is not a primitive $16$th root of unity, contrdicting our choice of $\zeta$. An analogous argument works for $\zeta^4$.