When $K$ is a local field with *finite* residue field $k$ of cardinality $q$, the maximal tamely ramified extension $M$ of $K$ contains the maximal unramified extension $N$ of $K$, the group $G=\mathrm{Gal}(N|K)$ (resp. $V=\mathrm{Gal}(M|N)$) is canonically isomorphic to $\hat{\mathbf{Z}}$ (resp. the group of roots of $1$ of order prime to $q$), and the action of the generator $\sigma\in G$ on $V$ is "raising to the power $q$".

The original sources for all this are Chapter 16 of Hasse's *Number Theory* and a paper by Albert (*On $p$-adic fields and rational division algebras,* Ann. Math. (2) 41 : 3 (July 1940), pp. 674–693). Iwasawa (*On Galois groups of local fields,*
Trans. Amer. Math. Soc. 80 (1955), 448–469) goes deeper into the question. These two papers are easily available on the Web.

As you might find the sources a little difficult to follow, a simplification and clarification has conveniently appeared a few days ago on the arXiv. The authors start from scratch and work out many illustrative examples. I hope you enjoy reading it.