# Inertia group in local class field theory

Let $$K$$ be a finite extension of $$Q_p$$ and $$H$$ is the inertia group of $$G_K$$, then we have the well-known local Artin isomorphism $$\widehat{K^\times}\cong G_K^{ab}$$ where the completion is for the norm groups. But we also have $$\widehat{K^\times}\cong O_K^*\times \hat{Z}$$, so we have the isomorphism $$O_K^*\times \hat{Z}\cong G_K^{ab}$$.

Question: Define the composition map $$H\hookrightarrow G_K\rightarrow G_K^{ab}\rightarrow O_K^*\times \hat{Z}$$ by $$f$$, then what is the image $$f(H)$$ in $$O_K^*\times \hat{Z}$$ ? Do we have $$f(H)\subseteq O_K^* \times \{1\}$$ ?

In the Artin map $$\newcommand{\Z}{\hat{\Bbb Z}}G_K\mapsto O_K^*\times\Z$$ the induced map $$G_K\to\Z$$ corresponds to the action on the maximal unramified extension $$H=K^{ur}$$. In detail we have $$G_K\to\text{Gal}(K^{ur}/K)\cong\text{Gal}(k^{alg}/k)\cong\Z$$ where $$k$$ is the residue class field. By definition the inertia group consists of the elements of $$G_K$$ acting trivially on $$K^{ur}$$, so the image of the inertia group is contained in $$O_K^*\times\{1\}$$. Indeed it is equal to $$O_K^*\times\{1\}$$.
In addition to your notations, let $$I_K$$ be the inertia subgroup of $${G_K}^{ab}$$. By definition, $$K^{ur}$$ is the fixed field of $$H$$ (resp. $$I_K$$) in $$G_K$$ (resp. $${G_K}^{ab}$$), so it is clear that $$f(H)=I_K$$, and we have only to show that the local reciprocity map $$\theta: K^* \to {G_K}^{ab}$$ induces an isomorphism $${O_K}^* \to I_K$$. It is classically known that the composition $$K^* \to {G_K}^{ab} \to G(K^{ur}/K) \cong \hat {\mathbf Z}$$ is the valuation map $$v$$, so we have a commutative diagram built from the exact sequences $$1\to {O_K}^*\to K^* \to {\mathbf Z}\to 0$$ and $$1\to I_K \to {G_K}^{ab} \to \hat {\mathbf Z} \to 0$$, with vertical connecting maps naturally induced by $$\theta$$ and Id. The induced map $$\theta :{O_K}^*\to I_K$$ is continuous, with dense image (this is because the reciprocity law in an unramified extension $$L/K$$ is explicitly given by $$x\to F^{v(x)}$$, where $$F$$ is the Frobenius element of $$G(L/K)$$). As $${O_K}^*$$ is compact, it follows that $$\theta$$ is surjective. The injectivity of $$\theta$$ is not trivial, it is one of the equivalent formulations of the so called existence theorem of local CFT.