Example of field extensions

Question

Assuming that for any n there is a Galois field extension with the Galois group isomorphic to $S_n$, I need to find an example of field extensions $F \subset G \subset E$ such that $G$ is normal over $F$, $E$ is normal over $G$, but $E$ is not normal over $F$. I was told to consider a larger field extension and then apply the fundamental theorem of Galois theory, but I don't see how and I don't see how there being a Galois extension with $n!$ elements fits in either. I thought this could be done with something like $F = \mathbb{Q}, G = \mathbb{Q}(\sqrt2), E=\mathbb{Q}(\sqrt2, 2^{1/4})$, but the hint is obviously directing me away from that

Answer 1

Your example seems good, so the hint seems unnecessary for you.

The way you'd go about using the hint would be to let $L$ be the Galois extension of $F$ with Galois group $S_n$, then find subgroups $\{e\} \subset K \subset H \subset S_n$ such that $K$ is normal in $H$ and $H$ is normal in $S_n$ but $K$ is not normal in $S_n$; the corresponding chain of fields $L \supset E \supset G \supset F$ would have $E$ normal over $G$ and $G$ normal over $F$ but $E$ not normal over $F$.