ABSTRACT
In the 1930s, German algebraists generalized this study by replacing the coefficient field C of the complex numbers with an arbitrary field when there could be no usual representation of the resulting algebraic function by a surface, Riemann or otherwise. The situation was especially involved when the coefficient field was of characteristic p (say, the field of integers modulo p). At the Notre Dame conference, I had lectured on certain lattices of such subfields. In Germany, Helmut Hasse and F. K. Schmidt wrote a long paper developing properties of such purely algebraic function fields: I spotted a subtle but substantial error in one of their proofs.
