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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.

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1 Answer 1

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If $K/L$ is an extension of fields, then the natural map $\operatorname{Gal}(K)^{ab} \to \operatorname{Gal}(L)^{ab}$ corresponds in class field theory to the norm map $K^\times \to L^\times$. So you just want to take the norm from $K$ to $\mathbb Q_p$ and compose it with the character you describe.

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