Ramified Coverings and Galois Theory

Abstract

The third chapter contains a geometric description of finite algebraic extensions of the field of meromorphic functions on a Riemann surface. For such surfaces, the geometry of ramified coverings and Galois theory are not only analogous but in fact very closely related to each other. This relationship is useful in both directions. On the one hand, Galois theory and Riemann’s existence theorem allow one to describe the field of functions on a ramified covering over a Riemann surface as a finite algebraic extension of the field of meromorphic functions on the Riemann surface. On the other hand, the geometry of ramified coverings together with Riemann’s existence theorem allows one to give a transparent description of algebraic extensions of the field of meromorphic functions over a Riemann surface.