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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 $n\geq 1$

??

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  • $\begingroup$ What's the question? $\endgroup$ – Lord Shark the Unknown Jan 18 at 4:50
  • $\begingroup$ I don't know why $F$ has non-zero characteristic $p$ $\endgroup$ – w.sdka Jan 18 at 4:57
  • $\begingroup$ As you say nothing about what $F$ is, then there is no reason to suppose it ha any given characteristic. $\endgroup$ – Lord Shark the Unknown Jan 18 at 4:58
  • $\begingroup$ 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. $\endgroup$ – Jyrki Lahtonen Jan 18 at 7:17

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