Classification of quasicharacters of a local field

Question

Let $F$ be a finite extension of $\mathbb{Q}_p$. There is a natural injection of abelian groups $\mathbb{Q}_p/\mathbb{Z}_p \rightarrow \mathbb{Q}/\mathbb{Z} \rightarrow \mathbb{R}/\mathbb{Z}$. Composing this map with the trace $F \rightarrow \mathbb{Q}_p$, we get a homomorphism of abelian groups $\lambda: F \rightarrow \mathbb{R}/\mathbb{Z}$. Let $S^1$ denote the unit circle in $\mathbb{C}^{\ast}$. It is known that if $\Psi: F \rightarrow S^1$ is a continuous homomorphism, then there exists a unique $y \in F$ such that

$$\Psi(x) = e^{2 \pi i \lambda(xy)}$$

In other words, the Pontryagin dual of $F$ is itself. If instead of continuous homomorphisms into $S^1$, we allow continuous homomorphisms into $\mathbb{C}^{\ast}$, is there also a nice classification?

Answer 1

All characters on $F$ are unitary: a compact subgroup of $\mathbb{C}^{\ast}$ is the same thing as a closed subgroup of $S^1$, the image of a compact subgroup of $F$ under a character is therefore contained in $S^1$, and $F$ is the union of the increasing chain of compact subgroups $\mathfrak p^{-1} \subseteq \mathfrak p^{-2} \subseteq \cdots$.

Answer 2

Remember, that the absolute value of a quasi-character is just a power of the absolute value on your field. So as usual you consider $\phi: F\to \Bbb C^*$ and consider $\phi/|\phi|:F\to S^1$ so as usual you can write any such as the product of a character and $|\cdot|^s$ for some $s\in\Bbb C$. See for example https://www.encyclopediaofmath.org/index.php/Quasi-character as an easy reference.