On the definition of algebraic closure

Question

Let $F$ be a field. By definition, the following are equivalent:

1. $F$ is algebraically closed.

2. Every nonconstant polynomial in $F[x]$ splits over $F$.

3. Every nonconstant polynomial in $F[x]$ has a root in $F$.

4. There is a subfield $K$, such that $F/K$ is algebraic and every nonconstant polynomial in $K[x]$ splits over $F$.

I want to know if the following condition also equivalent to the above:

(5) There is a subfield $K$, such that $F/K$ is algebraic and every nonconstant polynomial in $K[x]$ has a root in $F$?

If not, is there a counterexample?