It is well-known that the complex numbers $\mathbb{C}$ is a degree two extension of $\mathbb{R}$, where one possible minimal polynomial is $x^2 + 1$. Further, $\mathbb{C}$ is algebraically closed.
However, for non-archimedean completions of $\mathbb{Q}$, the story is quite different. Indeed, for any prime $p$, the algebraic closure of $\overline{\mathbb{Q}_p}$ is not a finite extension of $\mathbb{Q}_p$.
Other well-known examples include that the algebraic closure of the finite field $\mathbb{F}_q$ for any $q = p^k$, where $p$ is a prime, is not a finite extension. The algebraic closure of the rational numbers is not a finite extension of $\mathbb{Q}$.
So my question is: what are the fields $\mathbb{F}$ with the property that it is not algebraically closed but has a finite extension which is algebraically closed?
