Title says it all. Sorry my previous question was wrong; I see now; very stupid of me. So this is what I meant to ask. I am looking for a field extension of $\mathbb Q$, let's call it $K$, s.t. $K$ is a proper subset of $\bar{\mathbb{Q}}$, and $X^n-a$ has a root in $K$ whenever $a$ is in $K$.


Or is this question unknown?



Does there exist a proper intermediate field between ℚ and ℚ̅ closed under taking nth roots?
Yes because the algebraic closure of $\mathbb{Q}$ contains all roots of all polynomials. But what we learn in elementary Galois theory is that there are polynomials that cannot be solved using any finite number of applications of the operations of taking $n$th roots for every $n$ in $\mathbb{N}$ and field operations. See for example this old post on Math Stack Exchange, where this questions might be more suited for <https://math.stackexchange.com/questions/286944/quintic-polynomial-with-galois-group-a-5>


