If $E$ and $F$ are subfields of a finite field $K$ and $E\cong F$, prove that $E = F$.
A finite field is a simple extension of each of its subfields and $\mathbb{Z}_p$ is a subfield of every finite field. Hence $E\cong \mathbb{Z}_p(u)$ and $F\cong \mathbb{Z}_p(v)$ for some $u,v\in K$. Proving that $u = v$ given $\mathbb{Z}_p(u)\cong \mathbb{Z}_p(v)$ may be stronger than I need though, since $u$ and $v$ could be different generators for the same set.
Can anyone help me with this? Many thanks.
Edit: This is a Galois Theory free zone.