 
Summary: ARTICLE IN PRESS YJNTH:3622
Please cite this article in press as: Y. Aubry, R. Blache, On some questions related to the Gauss conjecture for function
fields, J. Number Theory (2008), doi:10.1016/j.jnt.2007.10.014
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On some questions related to the Gauss conjecture
for function fields
Yves Aubry a
, Régis Blache b,
a Institut de Mathématiques de Toulon, Université du Sud ToulonVar, France
b Equipe AOC, IUFM de Guadeloupe, Guadeloupe
Received 19 April 2007; revised 8 October 2007
Communicated by D. Wan
Abstract
We show that, for any finite field Fq, there exist infinitely many real quadratic function fields over Fq
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 Fq, of infinitely many real function fields over Fq
with ideal class number one (the socalled Gauss conjecture for function fields). We also show conditionally
the existence of infinitely many real quadratic function fields over Fq such that the numerator of their zeta
