Prove that if $K$ is a finite extension of F, $K \neq F$ then there exits $n\in \mathbb{N}$ such that $[K:F]=p^n$

Question

Let $K$ be a field of characteristic $p\neq 0$ , Let $F=\{a^p | a\in K\}$

Prove that if $K$ is a finite extension of F, $K \neq F$ then there exits $n\in \mathbb{N}$ such that $[K:F]=p^n$

I'm new to field theory the question tells us that $K\neq F$ but by Fermat we get that $a^p=a$ so thats mean that $K=F$ . What am I missing here? and can you give me a hint how to solve this?

Answer 1

Fermat says that $a^p = a$ in the field $\mathbb{F}_p,$ not every field of characteristic $p.$