Is a field of characteristic zero where -1 is a square algebraically closed?

Question

Let $F$ be a field of characteristic zero where $-1$ is a square. Must $F$ be algebraically closed?

Answer 1

Unless you give further assumption: no. Consider $F=\Bbb Q(i)$. This is a field extension of $\Bbb Q$ and thus of characteristic $0$ with $x^2=-1$ for $x=i$.