I am reading a textbook and I believe it makes a claim that is either false or not fully justified.
By the definition of $Q_{py}$ there is a tower $$\mathbb{Q} = L_0 ⊆ L_1 ⊆ \dots ⊆ L_n ⊇ \mathbb{Q}(\alpha)$$ such that $[L_{j+1} : L_j] = 2$ for $0 \le j \le n−1.$ Define $M_j = L_j \cap \mathbb{Q}(\alpha).$ Consider the $L_j$ and $M_j$ as vector spaces over $\mathbb{Q},$ and note that they are finite dimensional. We have $\dim L_{j+1} = 2 \dim L_j$ for all relevant $j.$ Therefore either $M_{j+1} = M_j$ or $\dim M_{j+1} = 2 \dim M_j.$
It suffices to prove $[M_{j+1} : M_j] < 3,$ or that $0 < 3\dim M_j - M_{j+1} = 3\dim (L_j \cap \mathbb{Q}(\alpha)) - L_{j+1} \cap \mathbb{Q}(\alpha),$ but I believe that this is false. For example, if $\dim M_1 = 1, n = 2,$ then $\dim M_2 = 4.$