Just as the characters of $\mathbb{Q}$ are in one-to-one correspondence with the adeles $\mathbb{A}_{\mathbb{Q}}$, which is the restricted product $\mathbb{R} \times \prod_{p\in\mathbb{P}}\mathbb{Q}_{p}$. Here restricted meaning that $(a_{\infty}, a_{2}, a_{3}, \ldots, ... )$ is in $\mathbb{A}_{\mathbb{Q}}$ if and only if $a_{p} \in\mathbb{Z}_{p}$ for all but finitely many $p \in\mathbb{P}$.
Does it make sense to ask about (continuous) characters of $\mathbb{Q}^{n}$? If yes, has it been studied? I would like to learn about it and requesting some references.