If $F\subseteq L\subseteq K$ are fields with $K/L$ and $L/F$ Galois, then $K/F$ is Galois?.
My intuition tells me that this is not true, but I can not find a counterexample, could someone give me one please? Thank you.
2026-03-29 17:33:58.1774805638
If $F\subseteq L\subseteq K$ are fields with $K/L$ and $L/F$ Galois, then $K/F$ is Galois?.
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Recall that if an extension $E/F$ is separable with $[E:F]=2$, then $E/F$ is also normal, hence is Galois. So try to find a non-Galois degree $4$ extension $K$ of $F=\mathbb{Q}$ which contains a subfield $L$ with $[L:F]=2$.