In Silverman's "The Arithmetic of Elliptic Curves", Cap. VII.4 (The Action of Inertia), we consider a local complete separable field $K$ with residue field $k$. Furthermore, we denote by $K^{nr}$ the maximal unramified extension of $K$.
Q: At page 194 Silverman claims without a proof or making a comment that the Galois groups $G_{K^{nr}/K}$ and $G_{\overline{k}/k}$ are equal. Recall, that $\overline{k}$ is the algebraic closure. Why is this identification true?
It is a standard fact of the theory of local fields that there is an equivalence of categories between the unramified finite extensions of $K$ and the finite separable extensions of the residue field $k$. Even the Wikipedia page https://en.wikipedia.org/wiki/Finite_extensions_of_local_fields mentions it. That identification of Galois group is an immediate consequence.