I've been having problems with the following proof:
Let $\alpha_1,\alpha_2,\alpha_3$ be real numbers such that $(\alpha_i)^2 \in Q$ for each $i$, and let $K=Q(\alpha_1,\alpha_2,\alpha_3).$ Show that the cube root of 2 is not contained in $K$.
I've tried a contradiction method, where I assume the cube root is contained, and then showing that you can't construct it with the alphas, and I've tried fiddling with the degrees of the extensions for a contradiction, but I can't get it to lead anywhere.