Gauss Problem for Function Fields

AbstractAccording to a celebrated conjecture of Gauss, there are infinitely many real quadratic fields whose ring of integers is principal. We recall this conjecture in the framework of global fields. If one removes any assumption on the degree, this leads to various related problems for which we give solutions; 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 least in some particular cases