Special linear group contains commutator subgroup of general linear group. I know that if $F$ field then $\det:GL_n(F)\to F^*$ with $F*$ (multiplicative group) non zero elements of $F$, $\det$ is a homomorphism with $\ker(\det)=SL_n(F)$.
Why commutator of $GL_n(F)$ is contained in $SL_n(F)$?

$\operatorname{det}:\operatorname{GL}_n(F) \rightarrow F^*$. So, if the kernel of this is $\operatorname{SL}_n(F)$, what is the quotient by this kernel a subgroup of? What does it say about the commutator of $\operatorname{GL}_n(F)$ if that quotient is abelian?
