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For the easiest case, assume that $L/E$ is Galois and $E/K$ is Galois. Under what conditions can we conclude that $L/K$ is Galois? I guess the general case can be a bit tricky, but are there some "sufficiently general" cases that are interesting and for which the question can be answered?

EDIT: Since Jyrki's reply seems to suggest that there is no general criterion on the groups. Can we say something if we put criterions on the fields? Assume say that $K=\mathbb{Q}$ or $K=\mathbb{Q}_p$?

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    What kind of criterion are you looking for? For example, assuming that the extensions in question are finite, there cannot be such a criterion in terms of the two Galois groups. This follows from the fact that given two finite groups $H$ and $K$, it is easy to construct a chain of finite groups $N_2\le N_1\le G$ such that $N_1$ is normal in $G$, $N_2$ is normal in $N_2$, $G/N_1\simeq H$, $N_1/N_2\simeq K$, but $N_2$ is not normal in $G$. Realize $G$ as a group of automorphisms of a field $F$, and let $L,E,K$ be the respective fixed fields of $N_2,N_1,G$. Then $L/K$ is not Galois.2011-12-04
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    @Jyrki: I guess the answer is then that no such conditions exist.2011-12-04

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