Let $p$ be a prime and $F$ a field with $p^2$ elements. Show that $F$ cannot have more than one proper subfield
What I have so far: Since $F$ is a field, $F$ is an abelian group under addition. Also, the number of elements in a proper subfield of $F$ must be $p$. How can I prove that this subfield is unique? Is this way the correct?
Hint: You're right that the proper subfield of $F$ must contain $p$ elements. It also must contain $1,$ and therefore $1+1, 1+1+1,$ etc.