# Is ture that $L/K$ is normal extension? [duplicate]

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?
• 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 $L/K$? – Bruce Jan 7 '18 at 9:56
No: consider e.g. $\mathbb{Q} \subset \mathbb{Q}(\sqrt[3]{2}) \subset \mathbb{Q}(\sqrt[3]{2}, \sqrt{5})$.