# Help proving the primitive roots of unity are dense in the unit circle.

I’m having difficulty understanding how to prove that the primitive roots of unity are in fact dense on the unit circle. I have the following so far:

The unit circle can be written $D=\{x\in\mathbb{C}:|x|=1\}$.
The set of primitive $m$-th roots of unity is $A_m=\{\zeta_k:\zeta_k^m=1,\zeta_k\text{ is primitive}\}$.

Hence, the set of all primitive roots $A$ is given by the union of $A_m$ over $m=1,2,3,\ldots$. But I can’t seem to get started on how to prove that $A$ is dense in $D$.

#### Solutions Collecting From Web of "Help proving the primitive roots of unity are dense in the unit circle."

To show the primitive roots of unity are dense in the unit circle, we must show that if we pick any point on the unit circle and any $\epsilon>0,$ there is some primitive root of unity with distance less than $\epsilon$ from the chosen point.

The easy way is to specialize $m$ to be prime – then every root of unity other than $1$ is primitive. Since the roots will be distributed evenly along the circle, if we pick a large enough prime we can make the distance between any two adjacent primitive roots less than $\epsilon.$ Then certainly if we pick any point on the unit circle, it will be within $\epsilon$ from a root.