Canonical homomorphism between $K/\mathcal O_K$ and $S^1$

Question

Is well know that there a isomorphism

1. $$\mathbb Q_p/\mathbb Z_p\approx \mathbb Z[1/p]/\mathbb Z\hookrightarrow \mathbb R/\mathbb Z\approx \mathbb S^1.$$

Now, let $K/\mathbb Q_p$ be a finite extension and let $\mathcal O_K$ be the ring of integers of $K$ i.e. $\mathcal O_K:=\{x\in K:\vert x\vert_p\leq 1\}$. And consider the additive quotient $K/\mathcal O_K$.

My question is if there a natural homomorphism as in the preceding case i.e. a homomorphism $K/\mathcal O_K\to \mathbb S^1$ such that if $K=\mathbb Q_p$ then the homomorphism $K/\mathcal O_K\to \mathbb S^1$ coincides with previous homomorphism $\mathbb Q_p/\mathbb Z_p\approx \mathbb Z[1/p]/\mathbb Z\hookrightarrow \mathbb R/\mathbb Z\approx \mathbb S^1$ ?

thanks you all.

I would appreciate any answer or reference.

Answer 1

$$O_K = \sum_{j=1}^n b_j \Bbb{Z}_p, \qquad K = \sum_{j=1}^n b_j \Bbb{Q}_p$$ where the $b_j$ have valuation $\in [0,1)$, $b_1=1$, $n=[O_K:\Bbb{Z}_p]$.

$$K/O_K= \sum_{j=1}^n (b_j \Bbb{Q}_p/b_j \Bbb{Z}_p)\cong (\Bbb{Q}_p/\Bbb{Z}_p)^n$$

Let $$f \in Hom(\Bbb{Q}_p/\Bbb{Z}_p,\Bbb{R/Z}),\qquad f(\frac{a}{p^k}+\Bbb{Z}_p)=\frac{a}{p^k}+\Bbb{Z},\qquad a\in \Bbb{Z}$$ Then $$c \to f(c.)$$ is an isomorphism $$\Bbb{Z}_p\to Hom(\Bbb{Q}_p/\Bbb{Z}_p,\Bbb{R/Z})$$ Whence $$Hom(K/O_K,\Bbb{R/Z})= \{ (\sum_{j=1}^n b_j A_j+O_K \to \sum_{j=1} f(c_j A_j)+\Bbb{Z}),c\in \Bbb{Z}_p^n\}$$ (where $A\in \Bbb{Q}_p^n$)

Once it is clear to you then for $p\nmid n$ there is some kind of canonical map, which is $$f \circ \frac1n Tr_{K/\Bbb{Q}}$$