Distinguishing Primitive vs. Nonprimitive Roots of Unity

Question

In a question here, the solution given states that $$\zeta=\cos{(\pi/8)}+i\sin{(\pi/8)}$$ is a primitive 8th root of unity. I was under the impression that the primitive roots of unity were given by $$\zeta_n=\cos{(2\pi/n)}+i\sin{(2\pi/n)}$$ which in case wouldn't it be a primitive 16th root of unity? I might have my language messed up here, so I was just looking for clarification.

Answer 1

Let me illustrate in case of $4$th roots of unity. Well $x^4=1$ has 4 roots $1, i, -i , -1$ and out of these four $i \; \mbox{and} -i$ have a special property that $i^k \neq 1$ if $k<4$ that means they are strictly fourth roots of unity and no smaller power of these numbers is 1.