# a question of local field

Let $K$ be a local field with mix char, $k$ residue field. We have an exact sequence

$0 \longrightarrow I \longrightarrow G_{K} \longrightarrow G_{k} \longrightarrow 0$

Then we obtain an action of $G_{k}$ on the abelization $I^{ab}$ of $I$. Let $T$ be the tame part of $I^{ab}$, and $F$ the frobinous element of $G_{k}$.

My question is

what is the action of $F$ on $T$?

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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.