I'm trying to prove that the subfields of a Galois Field $GF$ of order $p^n$ are isomorphic to a Galois field of order $p^r$ where $r|n$, and that there exists a unique subfield for each such $r$. I see that generally people use the Frobenius automorphism to prove this, but in my text I am not given any real relation between finite fields and the automorphism. I am given that any field $F$ has $p^n$ elements if and only if it is a splitting field for $f(t)=t^{p^n}-t$ over the prime subfield $\mathbf{Z}_p$. I'm also given the fact that for any $n\in \mathbb{N}$ and prime $p$ there is a unique finite field with $p^n$ elements.
From Lagrange's Theorem it follows that the order of any subfield of $GF$ divides $p^n$. If I could show that the order of any subfield of $GF$ must be $p^r$ where $r|n$ I think the rest would follow. I'm pretty sure all the ingredients are there but I just can't quite get the proof together. If anyone could help I'd be much obliged.