Characters and coverings of compact groups (Q465063)

Let \(G\) be a compact abelian group and let \(\hat{G}\) be its character group. The first result of the paper is a theorem that states if \(\chi_0\) is a character of \(G\) such that the equation \(x^n=\chi_0\) has a solution in \(C(G)\) (the space of continuous functions) then it has also a solution in \(\hat{G}\). The second result is a theorem that says that if all \(n\)-fold coverings of \(G\) are trivial then the character group is \(n\)-divisible, i.e. for each character \(\chi\) the equation \(x^n=\chi\) has a solution in \(G\).