Is $K=\overline{\Bbb Q}$?

Question

Let $K/\Bbb Q$ be an algebraic extension. Suppose $x^n-\alpha$ splits completely in $K$, that is, all its roots lie in $K$, for all $n\in\Bbb N$ and for all $\alpha\in K$. Is $K=\overline{\Bbb Q}$?

Answer 1

You're saying that $K$ is gotten from $\Bbb Q$ by adding all possible (complex) roots of numbers. This is not quite the algebraic closure of $\Bbb Q$, by the Abel-Ruffini theorem.

For instance, $x^5-x+1$ does not split over $K$