Gauss problem for function fields

According to a celebrated conjecture of Gauss, there are infinitely many re al quadratic fields whose ring of integers is principal. We recall this con jecture in the framework of global fields. If one removes any assumption on the degree, this leads to various related problems for which we give solut ions; namely, we prove that there are infinite families of principal rings of algebraic functions in positive characteristic, which are extensions of a given one, and with prescribed Galois, or ramification, properties, at le ast in some particular cases. (C) 2000 Academic Press.