I apologize if the title seems to be misleading as I couldn't conjure up a more relevant title. My question is that suppose we have that a prime $p$ and $q = p^k$ for some positive integer $k > 1$. Suppose we have $\mathbb{F}_q$. If $x \in \mathbb{F}_q$ has the property such that $x^p = x$, then $x^{p-1} = 1$ when $p$ does not divide $x$. Suppose $p$ does not divide $x$. So far, the order of $x$ divides $p-1$ by some theorem in algebra. My question is that since $x \in \mathbb{F}_q$, does that mean it lies in $\mathbb{F}_p$ since it has order $\leq p-1$? If so, could you refer me to a theorem that comments on that?
Thanks in advance