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Let $F$ be a field of characteristic $p$ and $E/F$ be a finitely generated field extension such that $E=F(E^p)$. Then show that $E/F$ is algebraic.

I have proven it in case $E$ is singly generated over $F$ but the general case seems difficult to prove.

Any help is deeply appreciated.

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    $\begingroup$ As a starting point, I would try looking at Saunders Mac Lane's "Modular Fields I". I don't know whether the answer to your question is there, but it contains a good deal of interesting information about separating transcendence bases that may help. $\endgroup$
    – Dave Benson
    Commented Aug 11, 2023 at 13:13
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    $\begingroup$ Thanks, will take a look at it! $\endgroup$
    – Akash Yadav
    Commented Aug 11, 2023 at 18:02

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