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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?

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  • $\begingroup$ Certainly not unless $|\alpha|_p=p^{nk}$ for some $k\in\mathbb Z$. $\endgroup$ Commented Jul 23, 2013 at 2:47
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    $\begingroup$ @AlexBecker $\alpha$ is a $p$-adic unit, which means that $|\alpha|_p = 1$. Regards. $\endgroup$ Commented Jul 23, 2013 at 2:51
  • $\begingroup$ Ah, missed that, sorry. $\endgroup$ Commented Jul 23, 2013 at 3:04
  • $\begingroup$ This is not my domain, but I am somewhat surprised that in the context of $\Bbb Q_p$, the term "unit" means neither (1) neutral element for multiplication, nor (2) invertible element (which would just be "nonzero" in $\Bbb Q_p$), but (3) element of absolute value$~1$. Long live ambiguity! By the way, if this is so, why not just talk about $p$-adic integers (where meaning (3) coincides with (2))? Clearly the $n$-th roots asked for are (unit) $p$-adic integers too. $\endgroup$ Commented Aug 2, 2013 at 6:40
  • $\begingroup$ @MarcvanLeeuwen It is customary in algebraic number theory that a unit of $\mathbb{Q}_p$ means an invertible element of the multiplicative monoid of the ring $\mathbb{Z}_p$. In a ring theory, a unit of a ring $R$ means an invertible element of the multiplicative monoid of $R$. en.wikipedia.org/wiki/Unit_(ring_theory) $\endgroup$ Commented Aug 2, 2013 at 7:04

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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.

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  • $\begingroup$ Nice! I was going to attempt an answer, but I misread the question and thought it was asking about the set $R$ consisting of all roots of $\alpha$. I think $R$ is finite iff $\alpha$ is a root of unity. I'm just not sure how to prove it without using the $p$-adic logarithm, a tool which I've glimpsed but not studied. Do you happen to know of a more elementary proof of this statement? $\endgroup$ Commented Jul 23, 2013 at 4:38

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