If a field $K$ (of characteristic 0) has no proper extensions of the form $K[\sqrt[n]{x}]$, is it algebraically closed?

Question

Let $K$ be a field, assume characteristic 0 if this simplifies things. Suppose we know that for any positive integer $n$ and any $x \in K$ the polynomial $X^n - x$ has a root in $K$. Is $K$ algebraically closed?

Working on the assumption that the answer is probably "no", how would one construct a counterexample?

Answer 1

The answer in characteristic $0$ is no. For instance, the "root closure"* of $\Bbb Q$ still wouldn't let you solve, say, an unsolvable quintic like $x^5-x-1 = 0$, because we know its solutions can't be written using roots in $\Bbb Q$, so it can't be written using roots in the root closure either.

*What I mean by "root closure" is this: Start with $\Bbb Q$, and consider it a subfield of $\Bbb C$ (or of $\overline{\Bbb Q}$). To $\Bbb Q$, add all $n$th roots of all elements of $\Bbb Q$. Then add all $n$'th roots of all elements in that new field, and so on. The root closure is the final result, i.e. the union of these fields. Any element in that field may be written with a finite expression involving only addition, subtraction, multiplication, division, roots and rational numbers.

Answer 2

In positive characteristic the situation is even worse. For instance, the field $K$ with two elements trivially satisfies your property, but the polynomial $x^{2} + x + 1$ has no roots in $K$.