Fix a field $\Bbb{F}$, and consider the following exact sequence: $$ 0\to~_n\Bbb{F}\to\Bbb{F}\to\Bbb{F}_n\to0 $$ where $~_n\Bbb{F}$ is the kernel of $\Bbb{F}\stackrel{\times n}{\longrightarrow}\Bbb{F}$ and $\Bbb{F}_n$ is the cokernel of this map.
I want to show the dual space of $\Bbb{F}_n$, that is $(\Bbb{F}_n)^*$ is isomorphic to $~_n\Bbb{F}$.
Could anyone give me some hint? Thanks in advance!
We have the right exact sequence $$F \xrightarrow{\cdot n} F \rightarrow F_n \rightarrow 0$$ and applying $\operatorname{Hom}(-,F)$ gives us the left exact sequence $$0\rightarrow \operatorname{Hom}(F_n,F) \rightarrow \operatorname{Hom}(F,F) \xrightarrow{(\cdot n)^\ast} \operatorname{Hom}(F,F)$$ Under the natural isomorphism $\operatorname{Hom}(F,F)\cong F$ the last arrow becomes $\cdot n$, hence by exactness of the second sequence $$\operatorname{Hom}(F_n,F)=\ker(\cdot n) = {}_nF.$$