The theorem of Riemann-Roch

Abstract

The classical theory of algebraic number-fields, as described above in Chapter V, rests upon the fact that such fields have a non-empty set of places, the infinite ones, singled out by intrinsic properties. It would be possible to develop an analogous theory for A-fields of characteristic p >1 by arbitrarily setting apart a finite number of places; this was the point of view adopted by Dedekind and Weber in the early stages of the theory. Whichever method is followed, the study of such fields leads very soon to results which cannot be properly understood without the use of concepts belonging to algebraic geometry; this lies outside the scope of this book. The results to be given here should be regarded chiefly as an illustration for the methods developed above and as an introduction to a more general theory.