Is there a cyclic dense subgroup of the multiplicative group $Z_p^*$?

Question

I know that, for $p$ prime, $(Z_p,+)$ is cyclic, in that the subgroup generated by 1 is dense in $(Z_p,+)$. Does there exist $x\in Z_p^*$ such that $\langle x\rangle$ is a dense subgroup?

If so, could we choose $x\in\mathbb{Q}$?

Answer 1

Let $p$ be odd. Then $(\Bbb Z/p^2\Bbb Z)^\times$ is cyclic. We can take a generator to be an integer $a$. Then $a$ also generates the cyclic group $(\Bbb Z/p^k\Bbb Z)^\times$ for all $k$. Therefore $a$ generates $\Bbb Z_p^\times$.

If $p=2$, then $\Bbb Z_2^\times/(1+8\Bbb Z_2)$ is non-cyclic of order $4$. Therefore no cyclic subgroup of $\Bbb Z_2^\times$ is dense.