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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 throughand the maximal unramified part through its Frobenius generator.

Choose the normalization in which a uniformizer maps to arithmetic Frobenius. For $x=p^{r}u$ with $r\in \mathbb{Z}$ and $u\in {\mathbb{Z}}_p^{\times }$, define ${\mathrm{Art}}_{{\mathbb{Q}}_p}(x)$ to act by ${\mathrm{Frob}}^r$ on the maximal unramified extension and byon 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 homomorphismis the Local Artin map; reversing both Frobenius conventions replaces the displayed inverse by the corresponding opposite normalization.

The translated cyclotomic polynomialis Eisenstein at $p$. Hence ${\zeta }_{p^n}-1$ generates a totally ramified extension of degreeEvery automorphism sends ${\zeta }_{p^n}$ to ${\zeta }_{p^n}^a$ for a unique $a\in (\mathbb{Z}/p^n\mathbb{Z}{)}^{\times }$. This gives an injectionand equality of orders makes it an isomorphism.

Normalize the Local Artin map so that $p$ maps to arithmetic Frobenius on the maximal unramified extension. For $u\in {\mathbb{Z}}_p^{\times }$, define its action on $K_n$ bywhere using $u$ rather than $u^{-1}$ gives the opposite Frobenius convention. These compatible maps, together with the image of $p$, define reciprocity on $p^{\mathbb{Z}}\times {\mathbb{Z}}_p^{\times }={\mathbb{Q}}_p^{\times }$ and hence on every finite abelian extension by the Local Kronecker-Weber theorem.