I have the following problem:
Let $E=\mathbb{Q}[\alpha, \beta]$ where $\alpha^2 \in \mathbb{Q}$, $\beta^2 \in \mathbb{Q}$, and $[E:\mathbb{Q}]=4$ If $\gamma \in E-\mathbb{Q}$ and $\gamma^2 \in \mathbb{Q}$ prove that $\gamma$ is a rational multiple of one of $\alpha, \beta, $or $\alpha \beta$.
I'm thinking a proof my contradiction would work, in combination with an argument concerning the degree of an extension, but it sounds fishy to me.