Decomposition of abelian closure of local field as $K^{ab }=K_{\pi}K^{unr} $ $K$ is a local field, we write $K^{\times}=U_K . \pi^{\mathbb Z} $ where $U_K$ is group of units of $\mathcal O_K $ and $\pi $ is the uniformizer of $K$.
I wish to prove that $K^{ab }=K_{\pi}K^{unr} $ where $K^{unr}$ is fixed field of $\phi _K(U_K) $  and $K_{\pi} $ is fixed field of $\phi _K(\pi) $.
From what I have gathered this is usually proved using Hasse-Arf theore, but can also be proven using the main theorems of local class field theory (see for example Milne's notes) which is what I wish to do.
But I could not find the proof anywhere.
Via the local Artin map, $\widehat {K^ \times} \cong U_K . \pi ^{\widehat {\mathbb Z}} \xrightarrow {\sim} \text{Gal}(K^{ab}/K) $. And it seems to me that the result should follow from this (the above equation looks similar to one we have to prove) but I cannnot seem to figure out how.
Feel free to give any reference and any help is appreciated.
 A: You didn't define $\phi_K$  but I guess it could be no other than the reciprocity map $K^* \to G_K^a:= Gal(K^{ab}/K)$. To go on we need to come back to the precise definition and properties of $\phi_K$, for which I think the best reference is Serre's book "Local Fields" ([LF] for short)
For a finite galois extension of local fields $L/K$ with group $G$, the consideration of $\hat {H^0}(G, K^*)$  allows to define an isomorphism $\theta:G^a \to K^*/NL^*$, where $N$ is the norm map, and its inverse is called the reciprocity iso., denoted $x \to (x, L/K)$ in [LF], Chap. XI, §3. Its functorial properties, listed at the end of §3, allow to define, by taking proj. lim. wrt to $L/K$ for a fixed $K$, the reciprocity morphism $\phi_K:K^*\to G_K^a, x \to (x, K^{ab}/K)$.
The next step is to put on $K^*$ a "normic topology" by taking the norm groups of finite extensions $L/K$ as a fundamental system of neighbourhoods of $1$. If the residual field of $K$ is finite (NB: in [LF], it is only supposed to be "quasi-finite" until chap. XIV), the so called existence theorem (XIV, §6) asserts that every closed subgroup of $K^*$ (wrt its natural topology) is a norm group from some finite $L/K$. Then, denoting by $I_K^a$ the inertia subgroup of $G_K^a$, it can be derived that the reciprocity map induces an isomorphism of topological groups $U_K \cong I_K^a$, where $U_K$ is the group of units of $K$ (op. cit. coroll. 2). On taking completions $(\hat .)$, the valuation map of $K^*$ straightforwardly produces an exact sequence of topological groups $1 \to U_K \to \hat{K^*} \to \hat {\mathbf Z} \to 1$. This means that, "loosely speaking,  $\hat{K^*}$ is obtained from $K^*$ by replacing $\mathbf Z$ by $\hat {\mathbf Z}$."
To speak less loosely, it remains to give a galois interpretation of $\hat {\mathbf Z}$. For this it will be convenient to refer to the chapter VI (written by Serre) of the Cassels-Fröhlich book "Algebraic Number Theory" ([CF] for short). Given a finite unramified extension $L/K$, for any $x\in K^*$ one has $(x, L/K)= F^{v(x)}$, where $F$ is the Frobenius of $L/K$ and $v(.)$ the normalized valuation of $K$ [CF], §2.5, prop. 2). This gives a commutative diagram linking the valuation exact sequence  $1 \to U_K \to K^* \to \mathbf Z \to 0$  to the inertia exact sequence $ 1\to I_K^a \to G_K^a \to \hat {\mathbf Z}\to 0$ (for diagrammatic details, see [CF], p. 144). This answers your question ./.
