Weierstrass factorizations in compact Riemann surfaces

Let V' be the complementary in a compact Riemann surface V of a point (or a finite set). In this paper are characterized the subfields, of the field of meromorphic functions in V', containing sufficient functions to verify a factorization property, similar to that of the classical Weierstrass theorem. It is also seen that the field generated by the Baker functions is not of this type, and the problem is solved of determining the divisors, in V', of the holomorphic functions admiting Weierstrass factorizations with Baker functions as factors. As an application, a theorem is obtained characterizing the infinite products, of meromorphic functions in V with bounded degree, which converge normally in V'.