Representations of $GL_n$

Question

Let $k$ be an algebraically closed field and comsider a homomorphism $f\colon \mathsf{SL}_n(k)\longrightarrow k^\times$. I know that every such homomorphism can be extended to all of $\mathsf{GL}_n(k)$ if $k=\mathbb{C}$. What about $k$ alg. closed?

Answer 1

$\operatorname{SL}_n(k)$ is its own derived group, so any such homomorphism is trivial.