Does $\sqrt[3^5-1]{3}$ exist in $\mathbb{Z}_3$ (or, $\mathbb{Q}_3)$?

I want to see whether $\sqrt[3^5-1]{3}$ exist in $\mathbb{Z}_3$ (or, $\mathbb{Q}_3)$, where $\mathbb{Z}_p$ is the ring of integers of the p-adic field.

If it exist then let $x=\sqrt[3^5-1]{3}$ which implies $x^{243}-3x=0$.

We have $x^{243} \equiv 0$ (mod $3$) but it's first derivative is not zero modulo $3$. So we can not use Hensel lemma

Is there other ways to see it ?

Solution 1:

Just extending comments to an answer:

No "proper" root of $p$, i.e. an element we would call $\sqrt[n]{p}$ for some $n \in \mathbb N_{\ge 2}$, exists in $\mathbb Q_p$.

That is for the simple reason that the $p$-adic valuation satisfies $v_p(x\cdot y)= v_p(x) +v_p(y)$, so that $v_p(\sqrt[n]{p})$ would need to be $\frac1n$; but $v_p(x) \in \mathbb Z$ for all $x \in \mathbb Q_p^*$.

Slightly more generally: If $a \in \mathbb Q_p$ is arbitrary, for the equation $X^n =a$ to have a solution in $\mathbb Q_p$ it is necessary that $n \vert v_p(a)$. (It is not sufficient in general.)