Let $L/K$ be an abelian extension and $\frak p$ a place of $K,$ Let $\theta_\frak p$$ :K_\frak p^*\twoheadrightarrow$$ G(\frak p)$ the $\frak p$-local Artin map (defined in Algebraic Number Fields by Gerald J. Janusz page 222-223 )
1)How to prove that $\theta_{\frak p} (O_{\frak p}^{\times})=T(\frak P/\frak p)$ where $\frak P$ a place of $L$ above $\frak p.$ and $T(\frak P/\frak p)$ the inertia group
2)How to prove that : $\mathrm{Ker}(\theta_{\frak p}|_{O_{\frak p}^{\times}})=N_{L_{\frak P}/K_\frak p}(O_{\frak P}^{\times}).$ ??
Edit : knowing that :$\mathrm{Ker}(\theta_{\frak p})=N_{L_{\frak P}/K_\frak p}(L_{\frak P}^{*})$ then we must prove that $\mathrm{Ker}(\theta_{\frak p}|_{O_{\frak p}^{\times}})=N_{L_{\frak P}/K_\frak p}(L_{\frak P}^{*})\cap O_{\frak p}^{\times}=N_{L_{\frak P}/K_\frak p}(O_{\frak P}^{\times}) $ for $\theta_{\frak p} (O_{\frak p}^{\times})=T(\frak P/\frak p)$ I showed that $\theta_{\frak p} (O_{\frak p}^{\times})\subset T(\frak P/\frak p)$ as follows : Let $\frak{n}=\frak{p}^a.\frak m$ where $\frak m$ be a modulus of $K$ such that the reciprocity law hold for $(L,K,\frak n)$ (assume $\frak p$ and $\frak m$ are relatively prime) in general the computation of $\theta_\frak{p}$$(x)$ for $x\in K^*_{\frak p}$ approximate the given $x$ by an $z\in K^*$ such that : $x\equiv^* z \pmod{\frak{ p}^a}$ because $K$ dens in $K_\frak p$ and by the approxiomation theorem there is $y\in K^*$ such that $y\equiv^* z \pmod{\frak{ p}^a}$ and $y\equiv^* 1 \pmod{\frak{m}}$ then $\theta_{\frak p}(x)=\varphi_{L/K}(\jmath_{\frak n}$$\imath(z))$ in fact $\theta _{\frak {p}}$ is defined as composed of $$\theta _{\frak p}\hspace{2mm} : \hspace{2mm} K_{\frak{p}}^{*} \longrightarrow\dfrac{K_{\frak {p}}^{*}}{U_{\frak p}^{(a)}}\longrightarrow\dfrac{K_{\frak{m},1}}{K_{\frak{m},1}\cap U_{\frak p}^{(a)}}\longrightarrow G(\frak p) $$ for more details see Algebraic Number Fields by Gerald J. Janusz page 222-223. Then if $u\in O_{\frak{p}}^{{\times}}$ there is $y\in K^*$ such that $y\in K^*$ and $y\equiv^* u \pmod{\frak{ p}^a},$ $y\equiv^* 1 \pmod{\frak{m}}$ then $x\in K_{\frak {m},1}$ and $\imath(x)$ is prime to $\frak p$ in particular $\imath(x)\in I^{\frak n}$, so $$\imath(x)\in I^{\frak n}\cap H^{\frak m}\subset H^{\frak n}=\mathrm{Ker}\varphi_{L/K}$$ $$\Rightarrow \theta_{\frak p}(u)=\varphi_{L/K}(\jmath_{\frak n}\imath(y))=1$$ then $\theta_{\frak p}(u)\in T$
thanks!!
There are different ways to approach the main theorems of local classfield theory.
Your additions/edits give the impression that you want to prove the local classfield theory results from global classfield theory. One place where this is carried out in some detail is S. Lang's "Algebraic Number Theory".
One might object to that style of argument because it does complicate the discussion of local classfield theory by entangling it with various approximation issues. The Artin-Tate notes on classfield theory, Neukirch's books/notes, and Milne's on-line notes (at http://www.jmilne.org/math/) all give more optimized proofs for local classfield theory.