Given a finite field $\mathbb{F}_p$, consider an extension of it; $\mathbb{F}_{p^m}$. If I'm given $\alpha \in \mathbb{F}_{p^m}$, then, if $\alpha^p = \alpha$, $\alpha \in \mathbb{F}_{p}$.
Why is this true?
2026-03-27 16:25:27.1774628727
When is an element of an extension field in the base field?
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The equation $x^p-x=0$ has at most $p$ roots. But it has $p$ roots within $\Bbb F_p$. So any root in an extension of $\Bbb F_p$ already lies in $\Bbb F_p$.