Let $p$ be an odd prime number. It is known that $\mathbb{Z}_p^{\times}$ is topologically cyclic.
Now let $\chi_{\mathrm{cyclo}} : Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p) \to \mathbb{Z}_p^{\times}$ be the $p$-adic cyclotomic character. This character factors through $Gal(\mathbb{Q}_p(\mu_{p^{\infty}} / \mathbb{Q}_p))$ which is also topologically cyclic.
Can I find a topological generator $g$ of $Gal(\mathbb{Q}_p(\mu_{p^{\infty}} / \mathbb{Q}_p))$ such that the image $\chi_{\mathrm{cyclo}}(g) \in \mathbb{N}$ ?
Best Answer
The cyclotomic characters induce an isomorphism between $Gal(\mathbb Q_p(\mu_{p^{\infty}})/\mathbb Q_p)$ and $\mathbb Z_p^{\times}$. So your question is equivalent to asking whether there is a topological generator of $\mathbb Z_p^{\times}$ which lies in $\mathbb N$.
The answer is yes: choose any natural number that is a primitive root modulo $p^2$ (and hence modulo $p^n$ for all $n$).
[This was already pointed out in an answer by user8268 that is now deleted.]