For a space \(X\), the ring of complex-valued continuous functions on \(X\) is denoted by \(C(X)\). Any \(n\)-fold cover \(\pi: E\to X\) induces a monomorphism \(\pi^*: C(X)\to C(E)\) so that \(C(E)\) is an algebra over \(C(X)\). Let \({\mathcal G}\) be the group of automorphisms of \(C(E)\) over \(C(X)\), i.e. of ring homomorphisms \(\phi: C(E)\to C(E)\) with \(\phi\pi^*=\pi^*\). Say that \(C(E)\) is a normal extension of \(C(X)\) provided that \(\{f\in C(E)/\forall\phi\in {\mathcal G},\quad \phi(f)=f\}=C(X)\). The paper develops a Galois theory relating subgroups of \(\mathcal G\) to intermediate covers of \(X\). In particular, if \(H\) is a finite index subgroup of \(\pi_ 1(X)\) and \(\pi: E\to X\) the corresponding cover, then \(H\) is a normal subgroup if and only if \(C(E)\) is a normal extension of \(C(X)\).
After further development of the theory it is applied to covers associated with Weierstraß polynomials over \(X\), i.e. maps \(P: X\times \mathbb{C}\to \mathbb{C}\) with \(P(x,z)=z^ n+\sum^ n_{i=1}a_ i(x)z^{n-i}\), where \(a_ i: X\to \mathbb{C}\) is continuous. If for each \(x\), \(P(x,z)\) has no multiple roots, then it gives rise to an \(n\)-fold cover \(\{(x,z)\in X\times\mathbb{C}/P(x,z)=0\}\to X\) by projection. Such polynomials split uniquely into irreducible factors whose Galois group action classifies the associated cover.