Let $L/K$ be an extension of number field. Let $Gal(L/K)$ is generated by $\sigma$.
Why Chebotarev's density theorem implies $ \exists {v \in Ω_L}$ such that $Gal(L_v/K_{v'})$ is generated by $\sigma^2$ ? Here, $L_v$ and $K_v'$ is completion of $L$ and $K'$ at $v$ and $v'$. $v'$ is restriction of $v$ to $K$.
Chebotarev's dentisty theorem I know is a Theorem $13.4$ of Neukirhi, algebraic number theory,chapter $Ⅶ$, which reads
Let $L/K$ be a Galois extension. Let $G$ be its Galois group. Then, for arbitrary $\sigma\in G$, there exists density of $P_{L/K}(\sigma)=\{I \in P(K)\mid$ I is unratified with $[\frac{L/K}{I}]=\sigma\}$, whose value is $\frac{\sharp〈\sigma〉}{\sharp{G}}$.
There may be another version of Chevotalef density theorem which fits in this context, but I'm wonder why this kind of global field statement implies titles statement(local field case).