Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 136 / 4 / b / Solution

Write $x={\pi }^{r}u$ with $r\in \mathbb{Z}$ and $u\in {\mathcal{O}}_K^{\times }$. If $x$ is an $m$th power, then $m$ divides $r$. Divisibility by infinitely many $m$ forces $r=0$, so $x$ is a unit.

Conversely, if $m$ is coprime to both the residue characteristic $p$ and $q-1$, exponentiation by $m$ is an automorphism on the finite residue-unit group and on every principal unit quotient; equivalently, use Hensel's lemma on $Y^m-u$. Hence every unit is an $m$th power for infinitely many such $m$. Thereforeas recorded by elements that are powers of infinitely many degrees in a local field.