Order Of The Intersection of Two Subfields.

Question

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 $E \cap F$ = $0$?

Answer 1

Extended hint:

So if $t\mid r$, and $t\mid s$, then, by the result you quoted, both $E$ and $F$ contain a unique copy of $GF(p^t)$. But also $GF(p^n)$ contains a unique copy of $GF(p^t)$, which means that the copies of $GF(p^t)$ inside $E$ (resp. $F$) must coincide with the one in $GF(p^n)$, and thus be contained in the intersection $E\cap F$.

In view of this the intersection $E\cap F$ is a copy of $GF(p^\ell)$, where $\ell=\gcd(r,s)$.