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If $L/K$ and $F/L$ are Galois extensions inside $\mathbb{C}$, must $F/K$ be a Galois extension?

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No, this is not true. –  Reader Oct 29 '12 at 23:01
    
Could you give a counter-example? –  Montez Oct 29 '12 at 23:05
    
I gave an example for if $L/F$ and $F/K$ are Galois then $L/K$ need not be Galois, but that is not what you are asking? –  Reader Oct 29 '12 at 23:35
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up vote 4 down vote accepted

Take the extension $$\mathbb{Q}\subset\mathbb{Q}(\sqrt{2}))\subset\mathbb{Q}(\sqrt[4]{2})$$ Then each of the intermediate steps are Galois as they are of degree two, but the total degree 4 extension $\mathbb{Q}\subset\mathbb{Q}(\sqrt[4]{2})$ is not Galois as some of the roots of the minimal polynomial of $\sqrt[4]{2}$ over $\mathbb{Q}$ is not in $\mathbb{Q}(\sqrt[4]{2})$.

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Consider the extension $\mathbb Q\subset\mathbb Q(\sqrt[4]{2})\subset \mathbb Q(\sqrt[4]{2},i) $. You have that $\mathbb Q(\sqrt[4]{2})/\mathbb Q$ is not Galois since it is not normal. Yu have to enlarge $\mathbb Q(\sqrt[4]{2})$ over $\mathbb Q$ in order to get Galois extension.

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But the question is: If $L/K$ and $F/L$ are Galois extensions inside $\mathbb{C}$, must $F/K$ be a Galois extension? Rather than whether a Galois extension can live inside a bigger Galois extension. –  Montez Oct 29 '12 at 23:11
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