Let $K$ be a number field and $v$ be it's one of $K$'s non-archimedian valuation. Then, I would like to prove $K_v(a^{1/m}) /K_v$ is unramified if only if $v(a)≡0 \pmod m$.
I know unramified extension of local field is in bijection with extensions of the residue field. Thus, the unramified extension is generated by roots of unity of order prime to the character of residue field of local field.
But I don't have tactics to judge given extension is unramified or not.
Thank you in advance.