Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 136 / 4 / c / 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 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.