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Let F be a char 0 field,
K be a normal extension of F
and L be a normal extension of K.
Can it be proved or disproved that L is normal extension of F ?

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up vote 7 down vote accepted

Let $F = \mathbb{Q}$, $K = \mathbb{Q}(\sqrt{2})$, $L = \mathbb{Q}(\sqrt[4]{2})$. Then $K/F, L/K$ are degree $2$ extensions, hence normal (and Galois) but $L/F$ is not normal (the splitting field of $x^4 - 2$ has degree $8$ over $\mathbb{Q}$).

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