On some questions related to the Gauss conjecture for function fields

We show that, for any finite field F-q, there exist infinitely many real quadratic function fields over F-q such that the numerator of their zeta function is a separable polynomial. As pointed out by Angles, this is a necessary condition for the existence, for any finite field F-q, of infinitely many real function fields over F-q with ideal class number one (the so-called Gauss conjecture for function fields). We also show conditionally the existence of infinitely many real quadratic function fields over F-q such that the numerator of their zeta function is an irreducible polynomial. (C) 2008 Elsevier Inc. All rights reserved.