Does a tower of Galois extensions in $\mathbb{C}$ give an overall Galois extension? If $L/K$ and $F/L$ are Galois extensions inside $\mathbb{C}$, must $F/K$ be a Galois extension?

Consider the extension $\mathbb Q\subset\mathbb Q(\sqrt[4]{2})\subset \mathbb Q(\sqrt[4]{2},i) $.  You have that $\mathbb Q(\sqrt[4]{2})/\mathbb Q$ is not  Galois since it is not normal. Yu have to enlarge $\mathbb Q(\sqrt[4]{2})$  over $\mathbb Q$ in order to get Galois extension.
