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Let $q$ be a power of a prime number, $K$ be a field containing $\textbf{F}_q$ and of transcendance degree one over it. Can I found $x\in K$ such that $K$ is a finite and separable field extension of $\textbf{F}_q(x)$?

Many thanks!

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This is possible if and only if $K$ is finitely generated over $\Bbb F_q$. This is the existence of a separating transcendence basis.

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