# Rational characters of a number field are powers of norm

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!).

If $$H$$ denotes the multiplicative group defined over $$K$$, then $$G=\mathrm{Res}_{K/\mathbb{Q}}H$$. By Section 2.61 of Milne's "Algebraic Groups - The Theory of Group Schemes of Finite Type over a Field", the group $$G_{\overline{\mathbb{Q}}}$$ obtained from $$G$$ by extension of scalars is isomorphic to the product of $$H_\sigma$$, where $$\sigma$$ runs through the embeddings $$K\hookrightarrow\overline{\mathbb{Q}}$$. It follows that the characters of $$G$$ defined over $$\overline{\mathbb{Q}}$$ are the maps $$\prod_\sigma f_\sigma^\sigma$$, where each $$f_\sigma$$ is a character of $$H$$ defined over $$\overline{\mathbb{Q}}$$. The characters of $$G$$ defined over $$\mathbb{Q}$$ are those maps $$\prod_\sigma f_\sigma^\sigma$$, which are fixed by $$\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$$. The condition "fixed by $$\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$$" means that $$f_\sigma$$ is independent of $$\sigma$$, and it is defined over $$K$$. In other words, $$f_\sigma$$ is the map $$x\mapsto x^k$$ on $$K^\times$$ with some $$k\in\mathbb{Z}$$ independent of $$\sigma$$, which means that $$\prod_\sigma f_\sigma^\sigma(x)=\prod_\sigma(x^k)^\sigma=N_{K/\mathbb{Q}}(x^k).$$