© 2004 by London Mathematical Society
© The London Mathematical Society
On the Norm Equation Over Function Fields
Institut für Mathematik, Universität Potsdam Postfach 601553, 14469 Potsdam, Germany, graeter{at}rz.uni-potsdam.de
Institut für Mathematik, Universität Potsdam Postfach 601553, 14469 Potsdam, Germany, weese{at}rz.uni-potsdam.de
Received 30 October 2002. Revision received 27 August 2003.
If K is an algebraic function field of one variable over an algebraically closed field k and F is a finite extension of K, then any element a of K can be written as a norm of some b in F by Tsen's theorem. All zeros and poles of a lead to zeros and poles of b, but in general additional zeros and poles occur. The paper shows how this number of additional zeros and poles of b can be restricted in terms of the genus of K, respectively F. If k is the field of all complex numbers, then we use Abel's theorem concerning the existence of meromorphic functions on a compact Riemann surface. From this, the general case of characteristic 0 can be derived by means of principles from model theory, since the theory of algebraically closed fields is model-complete. Some of these results also carry over to the case of characteristic p>0 using standard arguments from valuation theory.