This is a review problem for a Galois theory course.
Compute $[\mathbb Q(\zeta_8): \mathbb Q(\sqrt 2)]$ and find a basis. (Where $\zeta_8$ is the primitive $8$th root of unity).
I compute the degree as follows: Let $L= \mathbb Q(\zeta_8)$ and $K= \mathbb Q(\sqrt 2)$. Then $$[L:\mathbb Q] = [L:K][K:\mathbb Q].$$
It can be shown that $[L:\mathbb Q]=4$, since $\zeta_8$ is a root of the $8$th cyclotomic polynomial $x^4 +1 \in \mathbb Q[x]$, so it is algebraic over $\mathbb Q$, of degree $4$. Also, clearly $[K: \mathbb Q]=2$ because the minimal polynomial of $\sqrt 2$ is $x^2-2$.
Plugging these into our equation we get that $[L:K]=2$.
Now to find a basis, I'm a bit lost. I would like to find the minimal polynomial of $\zeta_8$ over $\mathbb Q(\sqrt 2)$, though I'm not sure how.
I know that a basis of $L/\mathbb Q$ is $\{1, \zeta_8, \zeta_8^2, \zeta_8^3\}$, and a basis of $K/ \mathbb Q$ is $\{1, \sqrt 2\}$, but I'm not sure about $L/K$.
My first instinct is to say it will be something like $\{1, \zeta_8 \sqrt{2} \}$, but I'm unsure about it.