Let $p$ be a prime number and $F$ a field of characteristic $p$. Let $\operatorname{Fr}_{p}: F \rightarrow F$ be the Frobenius map defined by $\operatorname{Fr}_{p}(x)=x^{p}$ for all $x \in F$.

(i) Prove that $\operatorname{Fr}_{p}$ is a field automorphism when $F$ is a finite field.

(ii) Is the same true for an arbitrary algebraic extension $F$ of $\mathbb{F}_{p}$ ? Justify your answer.

(iii) Let $F=\mathbb{F}_{p}\left(X_{1}, \ldots, X_{n}\right)$ be the rational function field in $n$ variables where $n \geqslant 1$ over $\mathbb{F}_{p}$. Determine the image of $\operatorname{Fr}_{p}: F \rightarrow F$, and show that $\operatorname{Fr}_{p}$ makes $F$ into an extension of degree $p^{n}$ over a subfield isomorphic to $F$. Is it a separable extension?