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**mylestone** $\displaystyle K$ is a finite field, $\displaystyle F$ is a subfield of $\displaystyle K$, and $\displaystyle m$ is a positive integer.

$\displaystyle L=\{a \in K \vert a^{p^m} \in F \}$.

Show that $\displaystyle L$ is a subfield of $\displaystyle K$ containing $\displaystyle F$. Moreover, show that $\displaystyle L=F$.

Easy enough to show $\displaystyle L$ is a subfield of $\displaystyle K$, and I have a proof that $\displaystyle L=F$ assuming that $\displaystyle L$ contains $\displaystyle F$, but I do not know why $\displaystyle L$ must contain $\displaystyle F$.