Power of Frobenius automorphism: Why $\phi(\alpha)^n = \alpha^{p^n}$ where $\phi$ is the Frobenius automorphism I cannot prove this. Even more, I have to conclude that $K = \{\alpha \in F | \phi^n(\alpha) = \alpha$} is a subfield of $F$.
What I have trying for the second part is:
$K$ is the splitting field of $p(x) = x^{p^n} - x$. So, $K \cong GF(p^n)$ that is a subfield of $F$. Is this correct?

A splitting field is a field extension of a field. Here $K$ is a subfield of $F$. So your argument can't work.
The arguments you should use:


*

*$\phi$ is a field automorphism. It would be useful that you try to understand why and look at the proof.

*Prove that the set of the elements fixed by a field automorphism form a subfield. It is not difficult if you write carefuly what it means.

