= Solution
The <Local Kronecker-Weber theorem> says that every finite abelian extension of $\mathbb Q_p$ is contained in a cyclotomic extension obtained by adjoining roots of unity. It identifies the totally ramified cyclotomic part through
$$
\operatorname{Gal}(\mathbb Q_p(\zeta_{p^\infty})/\mathbb Q_p)
\cong\mathbb Z_p^\times
$$
and the maximal unramified part through its <Frobenius> generator.
Choose the normalization in which a uniformizer maps to arithmetic Frobenius. For $x=p^ru$ with $r\in\mathbb Z$ and $u\in\mathbb Z_p^\times$, define $\operatorname{Art}_{\mathbb Q_p}(x)$ to act by $\operatorname{Frob}^r$ on the maximal unramified extension and by
$$
\zeta_{p^n}\longmapsto\zeta_{p^n}^{,u^{-1}}
$$
on every $p$-power root of unity. These compatible actions define an element of the abelian <Weil group>, because its residue action is an integral power of Frobenius. The resulting continuous homomorphism
$$
\operatorname{Art}_{\mathbb Q_p}:\mathbb Q_p^\times
\longrightarrow W(\mathbb Q_p^{\mathrm{ab}}/\mathbb Q_p)
$$
is the <Local Artin map>; reversing both Frobenius conventions replaces the displayed inverse by the corresponding opposite normalization.
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