Elements in an extension of a finite field $\mathbb{F}_p$.

Question

Assume that $\mathbb{F}_p$ is a finite field of $p$ elements and that we have the following extension $\mathbb{F}_p \subset\mathbb{F}$, where $\mathbb{F}$ is some infinite field (apparently of positive characteristic). Moreover, assume that we choose an arbitrary element $c \in \mathbb{F}$. If this element satisfies $c^p = c$, then does this imply that $c \in \mathbb{F}_p$?

Answer 1

Consider the polynomial $f(X)=X^p-X\in \mathbb{F}[X]$. Since $\mathbb{F}$ is a field, this polynomial has at most $p$ roots in $\mathbb{F}$.

On the other hand, every element of $\mathbb{F}_p$ is a root of $f$, so these $p$ elements must be precisely the roots of $f$ in $\mathbb{F}$. Hence if $c\in \mathbb{F}$ and $c^p=c$, then $c\in\mathbb{F}_p$.