# Separable polynomial and characteristic of field

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 $$n\geq 1$$

??

• What's the question? – Lord Shark the Unknown Jan 18 at 4:50
• I don't know why $F$ has non-zero characteristic $p$ – w.sdka Jan 18 at 4:57
• As you say nothing about what $F$ is, then there is no reason to suppose it ha any given characteristic. – Lord Shark the Unknown Jan 18 at 4:58
• For starters, the set of all roots of $f(x)$ is finite. If they form a field ... the rest follows pretty much immediately from standard proofs about the existence/uniqueness of finite fields. – Jyrki Lahtonen Jan 18 at 7:17