Let $E$ and $F$ be subfields of $GF(p^n)$. If $|E| = p^r$ and $|F| = p^s$, what is the order of $E\cap F$?
I read a corollary that "A finite field of order $p^n$ contains a unique subfield of order p^m for each $m$ | $n$ and no other subfields.
If that's the case, wouldn't that mean if two subfields have different orders, their intersection is $0$? So in this case, the order of $E \cap F$ = $0$?