Let $F$ be a non-archimedian local field with finite residue field. Let $\widehat{F}$ be the group of additive characters of $F$, that is the set of continuous group homomorphism $$\psi:(F,+)\rightarrow(\mathbb C^{\times},\times)$$ with the group structure given by complex multiplication of functions. I am interested in understanding the structure of $\widehat{F}$.
In the book "The local Langlands conjecture for $\operatorname{GL}(2)$" by Bushnell and Henniart, it is shown that any nontrivial character $\psi \in \widehat{F}$ gives rise to a group isomorphism $F \cong \widehat{F}$, by sending $a\in F$ to the character $a\psi:x\mapsto \psi(ax)$. However, no insight is given concerning the existence of such a nontrivial character in the first place.
Is there any way to argue that a nontrivial additive character of $F$ always exist? Is there a general way to construct one?
Thank you very much in advance.
Here is a general argument that shows that nontrivial additive characters exist for any nontrivial locally compact abelian group. By Pontryagin duality, letting $\widehat{G} = \operatorname{Hom}(G,S^1)$ we have that $G\cong \widehat{\widehat{G}}$ canonically. No nontrivial additive characters would imply that $\widehat{G}=1$, the trivial group, but the dual of the trivial group is the trivial group, so for any non-trivial locally compact abelian group there exist nontrivial characters. As any non-archimedean local field is a locally compact abelian group under addition, the result follows.
To demonstrate nontrivial characters, here's one defined on $\Bbb Q_p$. For $x\in\Bbb Q_p$, let $\{x\}_p$ be defined as the $p$-adic fractional part, that is, the sum of all the terms with negative power of $p$ in the usual $p$-adic expansion of $x$. Then $e^{2\pi i \{x\}_p}$ is a nontrivial additive character. To show this, we observe that $x-\{x\}_p\in\Bbb Z_p$, so $\{x\}_p+\{y\}_p-\{x+y\}_p$ is both a $p$-adic integer and a $p$-adic fraction, so it's an honest integer, and additivity follows.