I am trying to learn chapter IV, "Abstract class field theory", from Neukirch's book "Algebraic Number Theory". My problems occur in paragraphs 4 and 5 of that chapter. On two occasions, Neukirch claims that particular field extensions are Galois but I am not able to see why. Maybe someone can help.
In what follows, let $k$ be a field with separable closure $\overline{k}$ and absolute Galois group $G$. All other mentioned fields $K,L$ will be intermediate fields of $\overline{k}/k$, i.e. $k \subseteq K, L \subseteq \overline{k}$. Moreover, assume that a fixed, continuous and surjective homomorphism $d:G \to \widehat{\mathbb{Z}}$ is given. For an intermediate field $K$, denote by $G_K$ the Galois group $\text{Gal}(\overline{k}/K)$.
Defintions: For an intermediate field $L$, denote by $I_L$ the kernel of $d$ restricted to $G_L$. The maximal unramified extension of $L$ is the fixed field of $I_L$, i.e. the field $$\{a \in \overline{k}:\sigma a =a \ \ \forall\sigma \in I_L\}.$$ Moreover, an extension $L/K$ of intermediate fields is called unramified if $I_K=I_L$.
My Questions:
Let $L/K$ be a finite Galois extension and assume $(\widehat{\mathbb{Z}}:d(G_K)) < \infty$. Let $\widetilde{L}$ denote the maximal unramified extension of $L$. Neukirch claims that the extension $\widetilde{L}/K$ is also Galois. Why?
Let $L/K$ be a unramified finite extension. Neukirch seems to suggest that $L/K$ is Galois since he denotes the automorphism group by $G(L/K)$ - a notation he only uses in the case of Galois extensions and Galois groups. So, does $L/K$ being unramified and finite implies that $L/K$ is Galois? If so, why?