Open normal subgroups of Galois

Question

Let $K$ be a number field, $v$ a finite place of $K$. Let $K'$ be the maximal extension of $K$ unramified at $v$.

For instance, for $K = \mathbf{Q}$ and $v = p$ a prime, $K' = \mathbf{Q}(\{\zeta_{n}, p\nmid n\})$, $\zeta_n$ primitive $n$-th roots of unity.

Let $\overline{K}$ an algebraic closure of $K$. Is $\text{Gal}(\overline{K}/K')$ an open normal subgroup of $\text{Gal}(\overline{K}/K)$?

Answer 1

For profinite groups $G$, open subgroups $H$ always have finite index, because the map $G \to G/H$ is continuous, $G/H$ has the discrete topology, and $G$ is compact. In your question, this index $[K' :K]$ is infinite, so the answer is no.