On a Characterization of Compact Hausdorff Space $X$ for Which Certain Algebraic Equations Are Solvable in $C(X)$

Abstract

Let $X$ be a compact Hausdorff space and $C(X)$ the Banach algebra of all complex-valued continuous functions on $X$. We consider the following property of $C(X)$: for each $f \in C(X)$ there exist a $g \in C(X)$ and positive integers $p$ and $q$ such that $p$ does not divide $q$ and $f^{q} = g^{p}$. When $X$ is locally connected, we give a necessary and sufficient condition for $C(X)$ to have this property. We also give a characterization of a first-countable compact Hausdorff space $X$ for which $C(X)$ has the property above. As a corollary, we prove that if $X$ is locally connected, or first-countable, then $C(X)$ has the property above if and only if $C(X)$ is algebraically closed.