I'm trying to parse a page in Milne's CFT notes. The local reciprocity law gives us isomorphisms $\phi_{L/K}:K^\times/Nm(L^\times)\to \textrm{Gal}(L/K)$ for all abelian extensions $L$ of a nonarchimedean local field $K$. Taking limits we get by infinite Galois theory the isomorphism $\phi_K:\widehat{K}^\times\to \textrm{Gal}(K^{ab}/K).$
If we fix a uniformizer/prime element of $K$, call it $\pi$, then we have the canonical decomposition $K^\times \simeq \pi^\mathbb{Z}\times U_K,$ where $U_K$ are the units in the valuation ring of $K$. When we take limits, then we apparently get the decomposition $\widehat{K}^\times \simeq \pi^\widehat{\mathbb{Z}}\times U_K.$
My question is really how one should think about $\pi^\widehat{\mathbb{Z}}$. For example $\phi_K(\pi)$ acts as the Frobenius element on $K^{un}$ and it's easy to see how $\phi(\pi^a)$ acts for $a\in\mathbb{Z}$. My questions are the following:
How should one think of general exponents in $\widehat{\mathbb{Z}}$ instead of $\mathbb{Z}$?
How would one show that the fixed field of $\phi_K(\pi)$ i.e. $K_\pi$ is also fixed by $\pi^{\widehat{\mathbb{Z}}}$, so that infinite Galois theory gives us $K^{ab}=K_\pi\cdot K^{un}$?