Finding subfields of an Extension Field

Question

Let a $\in$ C be a root of the polynomial $X^4+ 1 \in Q[X]$ Consider the field extension Q(a) of Q.

Find three fields $K_1, K_2,K_3$ such that $Q \subset K_i \subset Q(a)$ for i=1,2,3.

I found out 2 fields namely, Q(i) and Q(√2) contained in Q(a). I couldn't find the third one. Iam not sure how to proceed further.

Answer 1

In general, if you have two extensions $\mathbb{Q}(\alpha)$ and $\mathbb{Q}(\beta)$, and you know they are different, then $\mathbb{Q}(\alpha\beta)$ is also different from the previous two.

Indeed, $\mathbb{Q}(\alpha\beta)\neq \mathbb{Q}(\alpha)$ since their compositum is $\mathbb{Q}(\alpha,\beta)\neq \mathbb{Q}(\alpha)$, and likewise $\mathbb{Q}(\alpha\beta)\neq \mathbb{Q}(\beta)$.