2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75036It has been shown by Madden that there are only finitely many quadratic extensions of k(x), k a finite field, in which the ideal class group has exponent two and the infinity place of k(x) ramifies. We give a characterization of such fields that allow us to find a full list of all such field extensions for future reference. In doing so we correct some errors in earlier published literature.To appear in the Journal of Number TheoryNumber Theory11R11; 11R29; 11R42text