On some questions related to the Gauss conjecture for function fields Original Research Article

We show that, for any finite field image, there exist infinitely many real quadratic function fields over image such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field image, of infinitely many real function fields over image 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 image such that the numerator of their zeta function is an irreducible polynomial.