The problem is found to be accessible from many standpoints. I regard as the most important key to the arithmetical part of this problem the general law of reciprocity for residues of
th powers within any given number field.
As to the function-theoretical part of the problem, the investigator in this attractive region will be guided by the remarkable analogies which are noticeable between the theory of algebraic functions of one variable and the theory of algebraic numbers. Hensel[27] has proposed and investigated the analogue in the theory of algebraic numbers to the development in power series of an algebraic function; and Landsberg[28] has treated the analogue of the Riemann-Roch theorem. The analogy between the deficiency of a Riemann surface and that of the class number of a field of numbers is also evident. Consider a Riemann surface of deficiency
(to touch on the simplest case only) and on the other hand a number field of class
. To the proof of the existence of an integral everywhere finite on the Riemann surface, corresponds the proof of the existence of an integer
in the number field such that the number