Let $k$ be a finite extension of $\mathbb{Q}\_p$ very often we use the isomorphism that $Gal(\overline{k}/k)^{ab} \simeq \hat{(k^{\times})}$ given by local class field theory.
My question would be do we have (and if yes how can I prove it) $ \hat{O\_k^{\times}} \times \hat{\mathbb{Z}} \simeq \hat{(k^{\times})}$ ? and is the image of the inertia subgroup of $Gal(\overline{k}/k)$ in $Gal(\overline{k}/k)^{ab}$ isomorphic to $ \hat{O\_k^{\times}}$ ?
I imagine that the proof of the first point can come from the fact that $\hat{k^{\times}} \simeq \widehat{O\_k^{\times} \times \mathbb{Z}}$ using the decomposition with an uniformizer. But then can I split the product ? Do we have $\widehat{O\_k^{\times}} = O^{\times}\_k$ ?



Decomposition of $\widehat{k^{\times}}$ occuring in local class field theory
Answering your questions: It is true that
$$\widehat{(k^{\times})} \simeq \widehat{O\_k^{\times} \times \mathbb{Z}} \simeq \widehat{O\_k^{\times}} \times \widehat{\mathbb{Z}} \simeq O\_k^{\times} \times \widehat{\mathbb{Z}}$$
as profinite groups, since


* $k^{\times}\simeq O\_k^{\times} \times \mathbb{Z}$ as groups by choosing an uniformizer $\pi$ of $k$: any non-zero element $a$ of $k$ can be written as $a=\pi^v b$, where $b\in O\_k^{\times}$ and $v\in \mathbb{Z}$ the valuation.
* Profinite completion commutes with products.
* $O\_k^{\times}$ is already profinite.


Secondly, it is also true that the image of the inertia group by the (local Artin) isomorphism is $O\_k^{\times}$. See for example [Poonen notes.](https://math.mit.edu/~poonen/papers/cft.pdf) 


