# Efficient way to find Galois group

In the book Abstract Algebra of Dummit and Foote, there is a problem as follows :

Let $K=\mathbb{Q}(\sqrt[8]{2},i), F_1=\mathbb{Q}(i), F_2=\mathbb{Q}(\sqrt{2}), F_3=\mathbb{Q}(\sqrt{-2})$. Prove that: $Gal(K/F_1)\cong Z_{8}, Gal(K/F_2)\cong D_8, Gal(K/F_3)\cong Q_8$

Here is my argument :