Genus fields of abelian extensions of rational congruence function fields

We give a construction of genus fields for congruence function fields. First we consider the cyclotomic function field case following the ideas of Leopoldt and then the general case. As applications we give explicitly the genus fields of Kummer, Artin–Schreier and cyclic p-extensions. Kummer extensions were obtained previously by G. Peng and Artin–Schreier extensions were obtained by S. Hu and Y. Li.