 Solution

A primitive eighth root of unity is ${\zeta }_8=(1+i)/\sqrt{2}$, and $\mathbb{Q}({\zeta }_8)=\mathbb{Q}(i,\sqrt{2})$. Indeed $i={\zeta }_8^2$ and $\sqrt{2}={\zeta }_8+{\zeta }_8^{-1}$, while the displayed expression gives the reverse containment. Since $\mathbb{Q}(\sqrt{2})\subset \mathbb{R}$, adjoining $i$ has degree two; adjoining $\sqrt{2}$ to $\mathbb{Q}$ also has degree two. By the tower law,