For $x$ lives in the algebraic closure of $\mathbb F_q$, $x^{q-1}=1$ if and only if $x \in \mathbb F_q$?

Question

Let $\mathbb F_q$ be a finite field of order $q$ and let $C$ be its algebraic closure. Taking $x\in C$, I wonder if $x^{q-1}=1$ if and only if $x \in \mathbb F_q$.

Of course $x \in \mathbb F_q$ implies that $x^{q-1}=1$. But the other direction is unclear to me.

Answer 1

The equation $z^{q-1}=1$ has precisely $q-1$ solutions in the algebraic closure $C$, which are $\mathbb F_q^*$. So $x^{q-1}=1$ implies that $x$ is one of those $q-1$ solutions. Sorry for this dump question :p