On the immersions between certain function fields on punctured compact Riemann surfaces (Q1114814)

Let \({\mathcal W}\) be a compact Riemann surface and \({\mathcal W}'\) be the complementary of a nonempty finite subset of \({\mathcal W}\). Let K(\({\mathcal W}')\) be the field generated by all meromorphic functions with finite divisor in \({\mathcal W}'\). In this paper it is proved that if the genus \(g_{{\mathcal W}}\) of \({\mathcal W}\) is \(>0\), then the restriction to the field \({\mathcal M}({\mathcal W})\) of meromorphic functions in \({\mathcal W}\) of every \({\mathbb{C}}\)-algebra isomorphism between any two fields of K(\({\mathcal W}')\) type must be an automorphism of \({\mathcal M}({\mathcal W})\). This is obtained as a consequence of a theorem asserting that the \({\mathbb{C}}\)-algebra immersions from \({\mathcal M}({\mathcal W})\) into a field K(\({\mathcal V}')\), corresponding to another compact Riemann surface \({\mathcal V}\), must take their values in \({\mathcal M}({\mathcal V})\) (if \(g_{{\mathcal W}}>0)\). As other consequences one obtains that K(\({\mathcal W}')\) and K(\({\mathcal V}')\) can only be isomorphic as \({\mathbb{C}}\)-algebras if \({\mathcal V}\) and \({\mathcal W}\) are isomorphic Riemann surfaces, and that every C-algebra immersion from K(\({\mathcal W}')\) into K(\({\mathcal V}')\) coincides with the composition of a map naturally induced by a nonconstant holomorphic map from \({\mathcal V}\) into \({\mathcal W}\), with an isomorphism of \({\mathcal M}({\mathcal V})\)-algebras.