Tower of Galois extension with some Properties.

Give an example of a tower of Field extension $\Bbb{Q}\subset F_1\subset F_2\subset F_3$ where $F_1/\Bbb{Q}$ and $F_3/\Bbb{Q}$ are Galois extensions but $F_2/\Bbb{Q}$ is not a Galois extension. Prove your assertion.? My attempt: $F_1=\Bbb{Q}$, $F_2=\Bbb{Q}(2^{1/3})$ and $F_3=\Bbb{Q}(2^{1/3},\zeta_3)$.

$F_2/\Bbb{Q}$ is not a Galois extension because it is not normal, $F_1/\Bbb{Q}$ and $F_3/\Bbb{Q}$ are Galois because any degree one extension is Galois and $F_3$ is splitting field of $x^3+2$?

• Does your source insist that $\subset$ means a proper subset? Conventions vary here. And this is kinda essential as in your example of $F_1=\Bbb{Q}$. – Jyrki Lahtonen Sep 11 '17 at 15:19
• If you want proper extensions at all the steps here is a hint: $D_4$ - the group of symmetries of a square. – Jyrki Lahtonen Sep 11 '17 at 15:20
• What do you know about subgroups of $D_4$? Which of them are normal, which are not? Also, you didn't comment on the meaning of $\subset$. Whether you need my hint depends on that meaning. – Jyrki Lahtonen Sep 12 '17 at 6:39