Consider the algebraic closure extension $\overline{\mathbb F_q}/\mathbb F_q$, where $q=p^m$. I wonder if the fixed field of the Frobenius automorphism is $$\sigma: \overline{\mathbb F_q} \to \overline{\mathbb F_q}, a\to a^q$$ is $\mathbb F^q$.
Clearly $\sigma$ fixed $\mathbb F^q$. But does it move any other elements in $\overline{\mathbb F_q}$?