Let $K$ be an extension of $\mathbb{Q}_p$. By local class field theory, the $p$-adic cyclotomic character $\mathrm{Gal}_K \rightarrow \mathbb{Z}_p^\times$ corresponds to a character $\chi : K^\times \rightarrow \mathbb{Z}_p^\times$.

Is there an explicit description of $\chi$?

When $K=\mathbb{Q}_p$, by Lubin-Tate's theory and using the formal group law of $\mathbb{G}_m$, I know that $\chi$ is the projection $\mathbb{Q}_p^\times = \mathbb{Z}_p^\times \times p^{\mathbb{Z}} \rightarrow \mathbb{Z}_p^\times$. I am looking for a similar description in the general (or special) case.