= Solution
Fix a non-Archimedean <local field> $K$, write $Q=|k_K|$, and use the \b[arithmetic Frobenius normalization] throughout: a <uniformizer> maps to the automorphism inducing $a\mapsto a^Q$ on the <maximal unramified extension>. The <Local Artin map>, or <local reciprocity map>, is the canonical continuous <group homomorphism>
$$
\operatorname{Art}_K:K^\times\longrightarrow\operatorname{Gal}(K^{\mathrm{ab}}/K),
$$
where $K^{\mathrm{ab}}$ is the <maximal abelian extension> inside a fixed <algebraic closure> with its separable part understood. Its finite projection to every finite abelian extension $L/K$ is onto and has kernel exactly the <norm subgroup of a local field extension>:
$$
\boxed{K^\times/N_{L/K}(L^\times)\xrightarrow{\sim}\operatorname{Gal}(L/K).}
$$
In particular, the quotient has order $[L:K]$. This is how <local class field theory> turns abelian extension problems into computations inside a multiplicative group.
One precise definition of the finite reciprocity map uses the <cup-product construction of local reciprocity>. For a finite <Galois extension> with group $G$, the <local fundamental class> $u_{L/K}\in H^2(G,L^\times)$ is specified by <local Brauer invariant> $1/[L:K]$. <Cup product> gives an isomorphism
$$
\widehat H^{-2}(G,\mathbb Z)=G^{\mathrm{ab}}
\xrightarrow{\ \smile u_{L/K}\ }
\widehat H^0(G,L^\times)=K^\times/N_{L/K}(L^\times).
$$
The finite <Local Artin map> is its inverse, composed with the quotient map from $K^\times$. For abelian $G$ the target is $G$ itself. Taking the compatible <inverse limit> over finite abelian extensions defines $\operatorname{Art}_K$.
The essential inputs to this cohomological construction have concrete arithmetic content. The <local Brauer invariant> identifies $\operatorname{Br}(K)$ with $\mathbb Q/\mathbb Z$; restriction to $L$ multiplies the invariant by $[L:K]$, so its kernel is cyclic of that order and has the distinguished generator $u_{L/K}$. For every subgroup $H\subseteq G$, <Hilbert theorem 90> gives $H^1(H,L^\times)=0$, while the same invariant calculation over the fixed field gives $H^2(H,L^\times)$ cyclic of order $|H|$. These are the hypotheses of the algebraic cup-product isomorphism in <Tate cohomology of a finite group>. It is obtained by <dimension shifting in group cohomology> with a <projective resolution> of the trivial module and the extension representing the fundamental class. Thus the norm-kernel statement comes from the degree-zero definition of <Tate cohomology of a finite group>, rather than from an arbitrary assignment of an automorphism to each element. In an <unramified extension> of degree $d$, the fundamental class is the <cyclic algebra> with <arithmetic Frobenius> generator and parameter a <uniformizer>; the isomorphism sends that Frobenius to the class of the uniformizer. This explains the chosen sign of reciprocity.
Several properties follow immediately from this construction. If $L\supseteq E\supseteq K$ are finite abelian extensions, restricting $\operatorname{Art}_{L/K}(a)$ to $E$ gives $\operatorname{Art}_{E/K}(a)$. The <Local Artin map> is therefore compatible with the finite projections defining its infinite target. For any finite separable extension $E/K$ and $a\in E^\times$, its norm compatibility is
$$
\operatorname{Art}_K(N_{E/K}a)=\operatorname{Art}_E(a)|_{K^{\mathrm{ab}}}.
$$
Here $K^{\mathrm{ab}}E/E$ is abelian, so the displayed restriction makes sense. An outline of the compatibility proof is to use the <corestriction map in group cohomology>: on multiplicative coefficients its degree-zero map is the norm. The cup-product projection formula, together with restriction and corestriction of the <local fundamental class>, gives exactly the commuting reciprocity diagram. The compatibility is an arithmetic instance of the functoriality of the cohomological definition.
The <Local Artin map> has dense image, because every finite abelian quotient receives a surjective map. Its kernel is trivial, and it extends to an isomorphism
$$
\boxed{\widehat{K^\times}\simeq\operatorname{Gal}(K^{\mathrm{ab}}/K),}
$$
where this <profinite completion> uses the open finite-index subgroups of the topological multiplicative group. The original map is generally not onto: on the unramified quotient it sends the integer valuation into $\widehat{\mathbb Z}$, rather than producing every profinite integer. Its units map onto the abelian <inertia group>. For a finite abelian extension, the corresponding unit image is its <inertia group>, while the valuation accounts for its residue extension. Indeed $v_K(Nz)=f(L/K)v_L(z)$, so the valuation quotient has order $f$; the full norm quotient has order $ef$, leaving unit image of order $e$. Since unit images act trivially on the residue field, they give precisely the <inertia group>. More finely, the image of the <higher principal-unit group> $1+\mathfrak m_K^r$ in a finite abelian extension is its rth upper <ramification group>, for each integer $r\geq1$. The use of <upper ramification numbering> is essential for compatibility with quotients.
The <existence theorem of local class field theory> states that every open finite-index subgroup $H\subseteq K^\times$ is $N_{L/K}(L^\times)$ for a unique finite abelian extension $L/K$. Inclusions are reversed: larger extensions have smaller norm subgroups. In the established reciprocity isomorphism, the proof is simply to take the fixed field of the open subgroup corresponding to $H$. Conversely a finite extension gives an open kernel of a finite continuous projection. Uniqueness follows because that kernel specifies its fixed field. The substantive existence problem is therefore to prove that the reciprocity isomorphism realizes all open finite-index subgroups; it is not just this final fixed-field argument.
The <Lubin–Tate description of local reciprocity> provides an explicit route to that existence problem. Choose a <uniformizer> $\pi$ and a <Lubin–Tate series> $f(T)=\pi T+T^Q$, with its associated <Lubin–Tate formal group> $F$. Solving $f(F(X,Y))=F(f(X),f(Y))$ successively by total degree constructs the law with linear part $X+Y$; the analogous equations construct the endomorphisms $[a]$ with linear coefficient $a$. The congruence $f(T)\equiv T^Q\pmod\pi$ supplies the divisibility needed at each step. These endomorphisms satisfy $[a][b]=[ab]$ and make the torsion an $\mathcal O_K$-module.
If $f_r$ denotes the rfold iterate, the primitive level-r <Lubin–Tate torsion> points are roots of
$$
\frac{f_r(T)}{f_{r-1}(T)}=\pi+f_{r-1}(T)^{Q-1}.
$$
Reduction modulo $\pi$ is $T^{Q^{r-1}(Q-1)}$, and the constant term is $\pi$, so this is an <Eisenstein polynomial>. Its roots therefore generate a <totally ramified extension> $K_{\pi,r}$ of degree $(Q-1)Q^{r-1}$. A primitive point generates the entire torsion module, and all points are its images under the integral convergent endomorphisms $[a]$. The torsion field is therefore a <Galois extension> of $K$. The action map into $(\mathcal O_K/\pi^r)^\times$ is injective, and the computed degree equals the order of this unit group, proving it is an isomorphism. Passing to the <inverse limit> gives $\operatorname{Gal}(K_\pi/K)\simeq\mathcal O_K^\times$ for the <Lubin–Tate tower> $K_\pi$.
The unramified tower and $K_\pi$ are <linearly disjoint field extensions>: a finite common subextension would be both unramified and totally ramified, hence trivial. The <Lubin–Tate description of local reciprocity> on their compositum is
$$
\operatorname{Art}_K(\pi^m u)|_{K^{\mathrm{nr}}}=\operatorname{Frob}_K^m,\qquad
\operatorname{Art}_K(\pi^m u)(\lambda)=[u^{-1}](\lambda)\quad(\lambda\in F[\pi^r]).
$$
The inverse on the unit action goes with the arithmetic Frobenius convention; changing to geometric Frobenius inverts the whole reciprocity map. The reciprocity theorem identifies this explicit action with the finite cup-product maps, independently of the chosen uniformizer. Its proof compares the fundamental class with the norm relations between consecutive torsion fields. The construction makes the possible finite kernels concrete: on the compositum of the degree-d unramified field with $K_{\pi,r}$ the kernel is
$$
\pi^{d\mathbb Z}(1+\pi^r\mathcal O_K).
$$
These subgroups are cofinal among open finite-index subgroups. Indeed any such $H$ contains $1+\pi^r\mathcal O_K$ for sufficiently large $r$, and contains $\pi^d$ for some positive $d$, since the class of $\pi$ in $K^\times/H$ has finite order. Taking the appropriate fixed subgroup in this explicit finite compositum realizes $H$. This proves the open-subgroup part of existence once the norm compatibility of the explicit and cohomological constructions is established. Together with the finite reciprocity maps it also shows that $K^{\mathrm{ab}}=K^{\mathrm{nr}}K_\pi$.
As a direct check, on a finite unramified extension the construction reduces to
$$
\operatorname{Art}_{L/K}(a)=\phi_{L/K}^{\,v_K(a)}.
$$
The earlier elementary unit-norm proof shows its kernel is exactly $\pi^{d\mathbb Z}\mathcal O_K^\times$. At $K=\mathbb Q_p$, the <Lubin–Tate cyclotomic example> takes $[p](T)=(1+T)^p-1$ and $F(X,Y)=X+Y+XY$. Its torsion is $\zeta-1$, so a unit $u$ acts by $\zeta\mapsto\zeta^{u^{-1}}$. These examples exhibit both the valuation and unit parts of local reciprocity with consistent signs.
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