On a characterization of the maximal ideal spaces of algebraically closed commutative -algebras

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