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 $x^3+2$?
Give your answers and also why did you thought of that answer?