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Give an example of fields $k\subseteq K\subseteq L $, and $l\subseteq L$, for which $l/k$ and $L/K$ are algebraic, $k$ is algebraically closed in $K$, and $lK = L$, but $l$ is not algebraically closed in $L$.

This question has already been posted here $k$ is algebraically closed in $K$, and $lK=L$ , but $l$ is not algebraically closed in $L$., but has not yet had an answer, the truth is that I can not find such an example, could someone please help me? Thank you very much.

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  • $\begingroup$ $l/k$ algebraic means $l \subset \overline{k} = \overline{l}$. So what did you mean ? $\endgroup$ – reuns Sep 20 '17 at 3:24
  • $\begingroup$ @reuns what else can I say about the problem? $\endgroup$ – user482152 Sep 20 '17 at 16:01
  • $\begingroup$ @reuns I have thought about taking $k =\mathbb{Q}, K =\mathbb{C} = L $ but I do not know how to choose $l$ so that the other is true, could you help me with this please? $\endgroup$ – user482152 Sep 20 '17 at 16:35
  • $\begingroup$ $l$ is algebraically closed in $\overline{k} \subset K$ $\endgroup$ – reuns Sep 20 '17 at 23:23

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