Pontryagin dual of the multiplicative group of a local field.

Question

Let $K$ be a local non-Archimedian field. Let $K^{\times}$ be the group of invertible elements of $K$.

Is there an explicit description of the Pontryagin dual of $K^{\times}$?

Answer 1

Assume first $k$ is a $p$-adic field. Then as topological groups $k^\times$ is isomorphic to $\mathbb{Z} \oplus \mathbb{Z} / (q-1) \oplus \mathbb{Z} / p^a \mathbb{Z} \oplus \mathbb{Z}_p^d$ for $a \geq 0$ and $d = [k : \mathbb{Q}_p]$. On the other hand, if $k$ has characteristic $p$, then $k^\times$ splits as $\mathbb{Z} \oplus \mathbb{Z} / (q-1) \oplus \mathbb{Z}_p^N$ for some $N$. This is Proposition II.5.7 of Neukirch's Algebraic Number Theory. It should thus suffice to consider each of these components. The only difficult part here is $\mathbb{Z}_p$, whose dual is the Prüfer group $\mathbb{Z}[ p^{-1}]/\mathbb{Z}\cong\mathbb{Q}_p/\mathbb{Z}_p$.