Suppose $f(x)$ is a monic separable polynomial over $F$ such that the set of all roots of $f(x)$ in the algebraic closure $\bar{F}$ is a subfield of $\bar{F}$.
Then, the splitting field of f(x) over $F$ is the subfield of $\bar{F}$. However, how to know the following properties
$ \textbf{ Question }$
(1) $F$ has non-zero characteristic $p$
(2) $f(x) = x^{p^n} - x $ for some $n\geq 1$
??