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Given a proper field extension $L/K$ (that is, $K$ can be considered as a proper sub-field of $L$). Can it still happen that $K\cong L$ via a field-isomorphism? I assume No, but I am utterly illiterate in basic field theory, so if this is really easy, then I would already be happy with a hint in the right direction.

M. Rumpy
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    Check Luroth theorem. If $K$ is any field and $x$ any indeterminate then any field $L$ with $K\subset L\subseteq K(x) $ is isomorphic to $K(x) $. – Paramanand Singh Dec 15 '21 at 12:16

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I suppose you can take $K$ to be a function field on countably many indeterminates, and get $L$ by adding one more indeterminate.

Andreas Caranti
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