On a $p$-adic unit and the existence of its $n$-th root

Question

Let $\mathbb{Q}_p$ be the field of $p$-adic numbers. Let $\alpha$ be a $p$-adic unit, i.e. an invertible element of the multiplicative monoid $\mathbb{Z}_p$. Consider the set $S = \{n \in \mathbb{Z}, n \gt 0\mid x^n = \alpha$ has a solution in $\mathbb{Q}_p\}$. Is $S$ an infinite set?

Answer 1

By Hensel's lemma, if $p\nmid n$ and $x^n\equiv \alpha\mod{p}$ has a solution $x\in\left(\mathbb{Z}/p\mathbb{Z}\right)^\times$, then $x^n=\alpha$ has a solution $x\in\mathbb{Z}_p$. Now, $\left(\mathbb{Z}/p\mathbb{Z}\right)^\times$ is cyclic of order $p-1$, so for any $n$ with $(n,p-1)=1$, $x^n\equiv\alpha\mod{p}$ has a solution $x\in\left(\mathbb{Z}/p\mathbb{Z}\right)^\times$. It follows that $S$ contains all $n>0$ with $(n,p(p-1))=1$, so $S$ is infinite.