Subfield has at most $p^n$ elements.

Question

Let $K$ be a field of characteristic $p$ and $n \in \mathbb{Z}_{>0}$. Consider the subfield $\{x \in K \ | \ x^{p^n} = x\}$, I need to show that this contains at most $p^n$ elements.

I use the Frobenius homomorfism $x \mapsto x^{p^n}$ to show it is indeed a subfield. But does someone have a hint on how to prove that the cardinality is at most $p^n$?

Answer 1

$X^{p^n}-X=0$ is an equation of degree $p^n$, and therefore cannot have more than $p^n$ roots.