On a characterization of the maximal ideal spaces of algebraically closed commutative $C^{\ast}$-algebras

Let $C(X)$ be the algebra of all complex-valued continuous functions on a compact Hausdorff space $X$. We say that $C(X)$ is algebraically closed if each monic polynomial equation over $C(X)$ has a continuous solution. We give a necessary and sufficient condition for $C(X)$ to be algebraically closed for a locally connected compact Hausdorff space $X$. In this case, it is proved that $C(X)$ is algebraically closed if each element of $C(X)$ is the square of another. We also give a characterization of a first-countable compact Hausdorff space $X$ such that $C(X)$ is algebraically closed.