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Let $F/K$ be a finite extension of fields. $S$ the set of subgroups of $\mathrm{Aut}_K(F)$ and $I$ the set of intermediate fields of the extension $F/K$. Define the function $\varphi:S\rightarrow I$ as $\varphi(G)= F^G$ where $F_G$ is the subfield of $F$ fixed by $G$. Could you help me to find an example of extension $F/K$ where $\varphi$ is not surjective?

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up vote 3 down vote accepted

$K=\mathbb{Q}$, $F=K(\sqrt[4]{2})$, $G$ is the group oforder two.

There are 2 subgroups and 3 intermediate fields ($K$, $F$, and $K(\sqrt{2})$).

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