Elements that are powers of infinitely many degrees in a local field
= Elements that are powers of infinitely many degrees in a local field
For a non-Archimedean local field $K$, an element of $K^\times$ is an $m$th power for infinitely many positive integers $m$ exactly when it belongs to $\mathcal O_K^\times$. Valuations prove necessity; for sufficiency, choose infinitely many $m$ coprime to the residue characteristic and to $|k^\times|$, for which exponentiation by $m$ is an automorphism of the unit group.