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This question already has an answer here:

Considering a field tower $K \subset E \subset L $, $E/K$ is separable extension,$L/E$ is normal extension, is ture that $L/K$ is normal extension?

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marked as duplicate by MatheinBoulomenos, Dietrich Burde, José Carlos Santos, John B, user99914 Jan 7 '18 at 16:20

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ I know that a tower of fields $K \subset E \subset L$,if $L/E$ is normal, $E/K$ is normal,but $L/K$ is possible not normal. But now what I want to ask is if $L/E$ is normal, $E/K$ is separable,how is $L/K$? $\endgroup$ – Bruce Jan 7 '18 at 9:56
  • $\begingroup$ The answer to the linked question adresses this question as well $\endgroup$ – MatheinBoulomenos Jan 7 '18 at 11:50
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No: consider e.g. $\mathbb{Q} \subset \mathbb{Q}(\sqrt[3]{2}) \subset \mathbb{Q}(\sqrt[3]{2}, \sqrt{5})$.

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