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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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1 Answer 1

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.

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