Find the radical extension of $\mathbb{Q}$ containing the number
$$\sqrt[4]{1+\sqrt{7}} - \sqrt[5]{2 + \sqrt{5}} $$
Wolfram is saying that the minimum polynomial is
$$x^4 + 12 x^3 - 42 x^2 - 468 x - 719$$
Definition of Radical Extension
A field $K$ is said to be a radical Extension of a field $F$
If $\exists $ a chain of fields
$$F =F_0 \subset F_1 \subset F_2 \subset \dots \subset F_t=K $$
such that $i=1, 2 , \dots ,t_t$ ,$F_t=F_{t-1}(u_i)$ and some power of $u_t $ is in $f_{t-1}$
The answer might look something like
$$\mathbb{Q} \subset \mathbb{Q} (\omega) \subset \dots \subset \mathbb{Q} (\omega, \sqrt[4]{1+\sqrt{7}},\sqrt[5]{2+\sqrt{5}} ) $$
Not sure what is the plan of attack. Do I find the minimal polynomial of the whole thing by hand. Can I split it into two? I am not sure how to do it without writing pages and pages of algebra.
Appreciate something to follow.