finitely generated subfield of algebraic closure of the finite field with $p$ elements Let $\mathbb{F}^{\operatorname{alg}}_p$ be the algebraic closure of the finite field with $p$ elements. 
I know that any finitely generated subfield of $\mathbb{F}^{\operatorname{alg}}_p $ is contained in some finite sub-union $\displaystyle{\bigcup_{n=1,2,\dots,N} \mathbb{F}_{p^n}}$.
However, I am lacking intuition for why this is so, and I'm having trouble conceiving what a finitely generated subfield of $\mathbb{F}^{\operatorname{alg}}_p $ would look look like. 
Though I will continue reading on the matter, any examples of such a finitely generated, and in turn finite, subfield would be greatly appreciated. 

For typographical simplicity, let’s write $\mathscr F\!_n$ for $\mathbb F_{p^n}$. Then one sees that $\mathscr F\!_m\subset \mathscr F\!_n$ if and only if $m|n$. It follows that $\mathscr F\!_m \mathscr F\!_n=\mathscr F\!_\ell$, where $\ell$ is the least common multiple of $m$ and $n$. Now, since every simply generated subfield of your algebraically closed field is a finite field $\mathscr F\!_m$ for some $m$, it follows that every finitely generated subfield is also simply generated.
