What are all the subfields of $\mathbb Q(p^{\frac{1}{4}}, q^{\frac{1}{2}})$ for any two distinct primes $p,q$?
From my intuition, I think that the degree $4$ subfields are $A=\mathbb Q(p^{\frac{1}{2}}, q^{\frac{1}{2}})$, $B=\mathbb Q(p^{\frac{1}{4}})$, and $C=\mathbb Q(q^{\frac{1}{2}}p^{\frac{1}{4}})$, and the degree $2$ subfields are $\mathbb Q(q^{\frac{1}{2}}p^{\frac{1}{2}})$ in $A$, $\mathbb Q(q^{\frac{1}{2}})$ in $A$, and $\mathbb Q(p^{\frac{1}{2}})$ in all $A, B, C$. But I don't know any rigorous way to prove that these are all the subfields and that there are no other subfields. I want to use Galois theory to find a corresponding Galois group but $\mathbb Q(p^{\frac{1}{4}}, q^{\frac{1}{2}})$ is not Galois. So I am not sure which procedure to take.