Consider a number field $K/\mathbb{Q}$ and the embedding of $K^* \hookrightarrow GL_n(\mathbb{Q})$. This is the set of rational points of a $\mathbb{Q}$-algebraic group $G \subseteq GL_n(\mathbb{C})$. Then is it true that any $\mathbb{Q}$-characters of $G$ will look like $g \mapsto \det(g)^k$ for some $k \in \mathbb{Z}$. That is, on $G_\mathbb{Q} = K^*$ the character will look like $x \mapsto N_\mathbb{Q}^K(x^k) $.

I heard someone make this remark in a discussion that all the rational characters on a number field are powers of norm. I have not been able to find a reference (or other non-norm characters!).