Find the field of intersection Let $\mathbb{F}_{p^{m}}$ and $\mathbb{F}_{p^{n}}$ be subfields of $\overline{Z}_p$ with $p^{m}$ and $p^{n}$ elements respectively. Find the field $\mathbb{F}_{p^{m}} \cap \mathbb{F}_{p^{n}}$. 
Could you give me some hints how I could show that??

Hints:
For all prime $\;p\;$ and $\;n\in\Bbb N\;$ :
$$\begin{align}&\bullet\;\; \Bbb F_{p^n}=\left\{\;\alpha\in\overline{\Bbb F_p}\;:\;\;\alpha^{p^n}-\alpha=0\;\right\}\\{}\\
&\bullet\;\; \Bbb F_{p^n}\le\Bbb F_{p^m}\iff n\mid m\;\;\end{align}$$
