Find the field of intersection

Question

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??

Answer 1

Often, when dealing with many field extensions, it is useful to draw a diagram depicting all of the fields that are/could be relevant and how they relate.

The case of finite extensions of finite fields is rather simple, however, since there is exactly one isomorphism class of field extensions of each degree (where the degree ranges over the positive integers).

In fact, the lattice consisting of all finite extensions of a finite field is isomorphic to the lattice of positive integers (ordered by divisibility), and translating questions from one lattice to another makes solving some problems very easy.

Answer 2

Hints:

For all prime $\;p\;$ and $\;n\in\Bbb N\;$ :

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