Let $\mathbb{K}$ be a local field.
Definition of local field:
Let $\mathbb{K}$ be a field and a topological space. Then $\mathbb{K}$ is called a locally compact field if both $\mathbb{K}^+$ and $\mathbb{K}^*$ are locally compact Abelian groups, where $\mathbb{K}^+$ and $\mathbb{K}^*$ denote the additive and multiplicative groups of $\mathbb{K}$, respectively.
a local field is a locally compact field with respect to a non-discrete topology.
How can I prove the existence of a non-trivial character $\chi$ on $\mathbb{K}$ ?