Frob_p under class field theory for cyclotomic extension of $\mathbb{Q}$

Question

Why does $$p^{-1} \in \mathbb{Q}_p^\times \subseteq \mathbb{A}^\times/\mathbb{Q}$$

go to $Frob_p$ for the cyclotomic extension $\mathbb{Q}(\mu_{p^\infty})/\mathbb{Q}$.

Answer 1

Oh, I see. the map. The isomorphism $\mathbb{A}^\times/\mathbb{Q} \to \mathbb{R}^\times_{\geq 0} \times \prod \mathbb{Z}_l^\times$ basically drops the "rational" part of an idele (because we are moding out by $\mathbb{Q}$ in $\mathbb{A}^\times/\mathbb{Q}$, and so $p^{-1} \in \mathbb{Q}_p$ goes to $(1, 1, ..., p^{-1}, 1, 1 ..) \equiv (p, p, \ldots, 1, p, ..)$ which goes to $p$ in $(\mathbb{Z}/f\mathbb{Z})^\times$ which is what $Frob$ goes to.